No
As we currently understand physical chemistry, all possible elements are known below whatever the state of the art number is now (Oganesson - Element 118).
The atomic number of an element (the number that determines which element it is) can only be an integer. It is, after all, the number of protons contained in the nucleus. Just like there are no unknown integers between 1 & 118, there are no unknown elements in this range.
Chemical reactions occur by complicated interactions between electron spin, electrons filling (or not) orbitals (valence electrons), and the charge of the atom (ionic bonds). All of this is due to the quantity of electrons (strongly influenced by the number of protons through the electromagnetic force) and how they fill the electron orbitals. So if you have a nucleus with no protons, it isn't "element zero," it is a neutron. Since neutrons have no charge, they do not bind any electrons. If a nucleus has no electrons, then it cannot interact chemically with anything.
Magic Stable Island
However, there's a current hypothesis called Stable Island. It posits that certain, as yet to be discovered, elemental isotopes will exhibit more stability than the elements around them in the periodic table.
Some theoretical calculations show that some element's isotopes in the magic "stable island" could have half lives as high as $10^9$ (a billion) years. More recent calculations indicate they would possess much shorter half-lives on the order of hours or days. Since no isotope of any elements supposed to be in the magic island have ever been observed, scientific consensus is leaning strongly towards the lower estimates.
The Stable Island is circled in the graphic. It looks like the center of the island is around atomic number of 112 and nucleon number of 276 & 278 (Copernicium - Cm). There is a second island of less stable nuclei at around Atomic Number 125 (nucleon number 294+). We've created some isotopes of Element 112, but its half-life is so short, we do not know its bulk properties. We haven't created any isotopes of Element 125:
Vertical axis is Atomic Number (number of $p$) and starts at 81 (Thallium). Each square represents an integer.
Horizontal axis is Nucleon Number (number of $p$ + $n$) and starts at around 205. Each square represents an integer.
Outlined boxes represent element isotopes already discovered or created.
Dashed black line shows the "optimal" $\frac{p}{n}$ ratio for a stable element isotope.
Finding an Unknown Element
I really like the idea of finding new elements, so let's assume that:
- The Stable Island exists
- There are undiscovered isotopes in one or more of these Stable Islands
- Some isotopes in the island have a half-life above $10^8$ years
- Supernovas make these elements
Plutonium (244Pu) has a half-life around $8 \cdot 10^7$ years. If we make some of it in a reactor, then for human purposes it sticks around forever. However, we have only once found any naturally occurring 244Pu (Do transuranic elements such as plutonium ever occur naturally?).
Just like Plutonium, a high-Z element with a half-life of $10^7$ - $10^8$ years would seem very stable to humans and exist for perhaps billions of years. All materials native to the solar system were create about 4.6 billions years ago in a supernova that cause the collapse of the dust cloud that formed our solar system. Since then radioactive elements have been decaying, so our proposed isotope would be short-lived enough that we should find very little or none of it in the materials native to our Solar System.
Interestingly, the Sun is sitting in a galactic feature called The Local Bubble. A series of supernovae which occurred from 10-50 million years ago blew the interstellar gases out of this region making the density of intergalactic gas in this region particularly low.
More importantly, the timing and location of these supernovae mean it is conceivable that a chunk of material from one of them could have made the trip to our solar system over the last 30-50 million years. Since these supernovae occurred only $10^7$ years ago, our proposed radioactive isotope should still have a very high percent of the original undecayed isotope remaining.
So imagine that humanity sees a body heading through the Solar System on a hyperbolic trajectory. This means that the body originated outside of our solar system and it will sail right through the Solar System unless we divert it. We would have to intercept and deflect it to pass near one of the gas giants to impart enough of a momentum change to capture it. Only one of these planets could deflect it enough to keep it in our Solar System. After deflection, we could find it is coated with a heavy elements (platinum group metals, uranium, plutonium, gold, and other materials that are rare on Earth).
What to use it for
Even if it burns more energetically with Oxygen than any other known chemical reaction (though physical chemistry suggests 118 is a Noble Gas, 112 will be close to a Noble Metal, and 125 will be a Rare Earth metal - so realistically you should expect any of these to react weakly or not at all with Oxygen), would humanity actually use it as a chemical fuel?
Certainly not.
It cannot be found on Earth, making it more valuable than any Terrestrial material you can think of (more valuable than Gold, Platinum, or even the most precious gemstones). If you were going to "burn" it, you wouldn't use the paltry energy release of chemical reactions (after all, how many people do you know who burn diamonds for heat?). Instead you'd go for the 1,000,000 $\times$ energy release of nuclear power.
Even so (and regardless of your "no nuclear" caveat), I suspect the material would be way too valuable to "burn" in fission reactors. With a $10^8$ish year half-life the only way we would get more would be if another such body flew through the Solar System. Don't expect it to be any sort of renewable energy supply. (If you wanted an SF analogy of the depletion of the Earth's fossil fuels, this might make for an interesting story though)
It would be used mostly for research - trying to discover just what the material could do. Or for the super wealthy, perhaps making some souvenir trinkets to wear (the material would be radioactive but not so radioactive as to be dangerous).
Edit 2/29/2016: So I was thinking about this and thought, hey what if we had a moderately large metallic asteroid on hyperbolic course through our solar system. We sent a probe to it and found it was chock full of elements from the stable island. If we could deflect its course, we'd have a huge quantity of the stuff available for all sorts of things (fission reactors, research, special material properties, etc.).
These materials would still not be used for their chemical reactions, the energy released would not be worth the energy investment to get the materials. It might be used for fission if it was a superior fission fuel (it would release more energy than the fission fuels we already use). Perhaps "burning" it would result in fission "ash" that were especially valuable elements (like platinum group metals), and released far fewer neutrons during the reaction. But that violates your no nuclear power criteria.
Regardless, this scenario also breaks your renewables scenario.
Other ways to get there
There are at least two other ways to get what you want.
Fusion
Some form of fusion, preferably cheap and low energy (aka "cold fusion") would serve nicely. There's no plausible and economic mechanism for this to work at the moment but it would not be a complete violation of physical laws to assume some way of doing it was discovered.
Metastable Helium
A Helium atom has 2 electrons. The lowest energy orbital is the "1S" orbital. The "1S" orbital can hold up to 2 electrons. However, those two electrons must possess different values of "spin" (one "up" and one "down"). If you instead give the Helium two electrons with the same spin value (e.g. two with the "up" spin), then one will sit in the "1s" orbital but its presence prevents the second electron from also falling into the "1s" orbital. Instead it sits in the "2s" orbital. This is called Metastable Helium.
Metastable Helium could conceivably provide energies far higher than any chemical reaction and give specific impulses of up to 10x that of $2\text{H}_{2\text{(L)}} + \text{O}_{2\text{(L)}} \rightarrow 2\text{H}_2\text{O}$. Since this is a "Real Thing", you wouldn't be violating the laws of physics to include it in your world.
Metastable Helium has a half-life of about 2.3 hours but it can be catalyzed to decay ("burn") faster.
The two main drawbacks of Metastable Helium are:
- It is not a fuel, it is an energy storage mechanism (you still need
power plants to make your energy).
- Metastable Helium is, well, metastable. It has a tendency to
spontaneously release its energy. Helium switching from metastable
to stable tends to catalyze surrounding Metastable Helium to do the
same thing. If you have a large fuel tank, the large quantity and
the high energy density of the substance tends to lead to an "Earth
shattering Ka-Boom"
There's a variant of Metastable Helium that reduces of some of its problems (e.g. making it more stable and giving it a longer half-life). This is called diatomic metastable Helium. You bond a metastable Helium to a stable Helium then chill it until it forms a solid. The resulting material has a half-life measured in years but it releases its energy when exposed to heat.
Unfortunately, this halves the energy density of Metastable Helium - but that's still much better than typical chemical reactions.
Nuclear Isomers
Another possibility is a nuclear isomer.
Imagine a small but very elastic balloon with a wide mouth. Fill this balloon with 72 black ping pong balls (protons - $p$) and 102 white ping pong balls (neutrons - $n$) representative of the nucleus of Hafnium (Hf). Both $p$ and $n$ are nucleons. Each ping pong ball has a Mexican jumping bean in it. Shake your balloon until you get the minimum possible surface area - this is known as your minimum or ground energy state. Now carefully pull one nucleon out of its ground state and move it to the other side of the balloon so it sticks out. This is a nuclear isomer and represents the "excited state" of the nucleus.
If you just let the configuration sit for a while, the random energy supplied by the Mexican jumping beans will eventually cause the ping pong balls to suddenly shift back into their ground state. This will release a sound ("voomp!"). The balloon represents a nucleus. The ping pong balls represents the nucleons. The sound is the gamma ray ($\gamma$) released when the nuclear isomer releases its energy.
If you bang the balloon sufficiently hard, the nucleus will reconfigure to the ground state too. This is the what scientists are trying to prove in the lab.
While Metastable Helium uses excited electron states to store energy, nuclear isomers store energy in excited nucleons. Just as nuclear reactions are $10^6 \times$ more powerful than chemical ones, excited nucleons can store enormously more energy than electrons (on the order of $5 \cdot 10^5 \times$ more than most chemical reactions).
Similar to how the range of half-lives for radioactive elements range from picoseconds to more than tens of billions of years, the same is true for nuclear isomers. It is thought one isomer has a half-life of around $10^{15}$ years - it has never been observed to decay and has no practical use. Others have half-lives so short they are of no practical use either because all atoms in the sample appear to spontaneously decay. However, Hafnium has a nuclear isomer with a very convenient half-life of 31 years and might be useful.
There are a few problems with nuclear isomers. These include an inability to trigger the release of the energy (**more on this below). Energy is released from Hafnium as gamma rays (which requires shielding). As with Metastable Helium, nuclear isomers are not a fuel but are an energy storage mechanism. They are only renewable in the same way that batteries (and Metastable Helium) are renewable - they can be recharged and reused.
There has been some controversial research indicating that a method of stimulating the release of energy from nuclear isomers has been found. So far though, the amount of energy required to stimulate them is more than the energy they can release.
Despite all those issues, nuclear isomers are theoretically possible energy storage mechanism and they could become a component of an awesome energy storage infrastructure. You would charge them up and then use them like batteries whose power slowly wound down.
Energy Densities
You won't find any non-nuclear energy source with energy densities greater than those of nuclear. It has to do with the type of forces involved (Nuclear Strong and Weak versus Electromagnetism) and is fundamental to the nature of the Universe.
Many people don't have an innate understanding of the relative magnitudes of the energy released between chemical and nuclear power. So let's use distance as representative of energy. If 1 cm correlates to the energy released by the most powerful chemical reactions, then 16 kilometers is the energy released by nuclear fission. Fusion is 100+$\times$ more powerful (represented by a distance 1,600 kilometers). Generating power using chemical energy is just not very effective compared to nuclear.
This is the relative energy densities of several materials:
- Antimatter > $12 \times$ Fusion (Wiki value of $150 \times$)
- Fusion > $4 \times$ Fission (Wiki value of $100 \times$)
- Fission > $10^6 \times$ chemical (Wiki value of $1.6 \cdot 10^6 \times$)
- Nuclear Isomer > $5 \cdot 10^5 \times$ chemical
- Metastable Helium > $10-100 \times$ most other chemical (Wiki value $10 \times$)
- Diatomic Metastable Helium > $5-50 \times$ most other chemical (Wiki value $5 \times$)
- Most chemical fuels > $100 \times$ most renewables (direct comparison
is difficult because renewable fuels are often "free" but their
infrastructure is huge and costly)
Incidentally, you get the most energetic chemical reactions between elements by combining elements from the upper right of the periodic table (oxidizers like Fluorine/Oxygen) with those on the lower left (reducers / alkali metals). An as yet not created alkali metal Ununennium - Element 119 would satisfy your requirements. However, this element is not expected to have a half-life greater than microseconds and wouldn't survive a trip from the nearest supernova (the stellar event that creates such elements) to the Solar System.
You can create molecules with more combustion energy by many mechanisms. Most of these extremely powerful explosives are not stable and therefore not safe to use in most cases. Others (e.g. Octanitrocubane aka Cubane), are so difficult to create that they are too expensive to manufacture in quantity.
Octanitrocubane (molecular formula: C8(NO2)8) is a high explosive
that, like TNT, is shock-insensitive (not readily detonated by shock).
...
Octanitrocubane is thought to have 20–25% greater performance than
HMX (octogen).
But ultimately chemical reaction energies are limited by the energies available by chemical bond strengths.
Renewables
Renewables at least as commonly conceived (e.g. wind & solar) possess extremely low energy densities. Consider a wind farm with 1000 of the large 1 MW wind turbines. It would cover many square miles and generate 1/5 or less of the energy produced by a single mid-sized 1 GW coal burning or nuclear plant which occupies a just a couple of acres of land (actual power generated and not installed theoretical capacity).
If you need energy density, then you need nuclear.
If you want renewables, then you have to live with extremely poor energy densities.
If you want both, then you need something like the Metastable Helium (see above) to store the energy produced by your low energy density power plant in a form that has high energy density (but you still have to live with the huge, low energy density wind farm).
Although it is currently popular to extol the virtues of "renewables", ultimately renewables come from sunlight and sunlight comes from the Sun which is a giant nuclear power plant. Why deal with all the middlemen (intervening processes) each of which has fairly steep efficiency loss. Why not work directly with the nuclear power?
How might this work?
Using the diatomic metastable Helium example...
Suppose scientists were examining cometary materials (how they were collected isn't important) and discovered one such compound contained metastable helium in some form that was much more stable than any we've discovered so far. Just knowing that a material like that existed and having some examples would lead to a huge new area of research.
Eventually, everything might run on the new energy storage media using the metastable helium compound as the battery and huge renewables spread across the planet (or the Moon) to power up those batteries.
We would not depend upon mining the materials from heavenly sources.
As an alternative if it is important in your story that we had to mine the fuel source, then you could always turn to mining the Moon and other astronomical bodies for 3He. That requires you to use nuclear though. On the plus side, 3He throws off significantly fewer neutrons than most other fusion reactions.