Maximizing short-term profits will be different than long-term profits. A rare element synthesized will lose value faster as the relative amount put into the market will depress the price faster as opposed so something less valuable but more common. It will probably involve some speculation and calculations of what is in demand and what you can get the best prices for.
The ideas below are likely not the most valuable, but they have low (or no) materials costs.
For reference, first is a list of what the device is observed to be able of, and as such, what the rest of the answers are based on.
What we know about the Device:
- Any two atoms of a substance can fuse into a single output element comprising the entirety of the matter (22H -> 4He and 216O -> 32S)
- Two different atoms can be fused into a single atom containing the entirety of the matter present (2H + 16O -> 18F )
- Only one new product can be made in a reaction -- no side products are created. That or any side products that are created when molecules of the original product are split. Where the hydrogen came from the Oxygen/Sulphur reaction in the story is a bit nebulous.
- The Transmuter cannot fuse protons and electrons into a neutron, ass evident by the starting product for helium being the rarer deuterium as opposed to the more common protium. It is also indirectly stated in the question itself.
- We do not know the energy cost of elemental synthesis so ideally what we make should take a single step wherever possible.
- It is stated that any reaction is possible, which means it could be possible for more than two atoms to be involved in a reaction, though that is not explicitly stated
- It is unknown whether or not it transmutes an entire element or just a single isotope of an element.
This suggests that the limitation of what we can actually do is the neutron count of the atoms we want to make. While the lighter elements have a lower proton to neutron ratio, the higher up the periodic table we go the more neutrons we need in order to create a stable atom.
So our unwritten cost limitation is how cheap can we get elements with a higher neutron to proton ratio to get the capacity to synthesize the heavier elements.
Since we can't do the reaction 1H2 -> 21n as that violated the rules stated in the question regarding conservation of subatomic particles, it is actually a bit trickier than it looks as we can't just create what we feel like by zapping enough hydrogen into neutrons to solve our problems.
Easy Small Change
The atmosphere we breathe is primarily made up of nitrogen and oxygen, of which the primary isotopes are 14N and 16O. From those two, you can easily make Silicon, Phosphorus, and Sulphur by fusing the main components of air together. Silicon-28, Phosphorus-31 and Sulphur-32 are stable isotopes so you are likely sticking with Silicon or Sulphur for that one.
Your input materials are essentially free (atmosphere) so there is no worry about pesky questions about why one needs a large quantity of element X. Plus, your detritus of this is some noble gasses, carbon dioxide, and methane.
The downside is that this does not necessarily maximize profits by creating the most expensive thing. But this is something you can do in secret with only your energy bills and a large pile of silicon/sulphur to give you away. However, if the energy costs are too much, you won't actually be making a profit.
Nickel and Diming
Iron is cheap, and helium is not terribly expensive either. Nickel-60 is thus not an outrageous thing to create and can sell for a good price compared to the base cost of the materials. The primary reaction is 56Fe + 4He -> 60Ni.
Your primary benefit to this is that the other natural isotopes of iron, Iron-54, Iron-57, and Iron-58 also make stable isotopes of nickel through this process which means you do not care about what isotopes of iron are in your ingot, just that it is pure iron.
Again, you aren't likely making mad money doing this, but it is something you can do with little additional people or resources.
This previous answer mentioned Rhodium at a rather impressive selling price. I do not know if this is the most cost-efficient way to do it by the rules as I understand them, but you can transmute it using Krypton and Fluorine in the following: 84Kr + 19F -> 103Rh.
While there is only the one stable isotope of fluorine (19F), the krypton is a minor issue as the most common isotope is only a bit over 50% of the naturally occurring krypton.
A brief bit of research shows that the other isotopes of krypton will create rhodium isotopes that decays into either palladium or ruthenium.
Humourous Mad Science/Business
If you can crack molecules of something apart for your reactions, one can take the generally unwanted gas carbon dioxide, and fuse the carbon-12 atoms into magnesium-24 and have oxygen as your byproduct. Well, the magnesium might oxidize first, but you are still basically converting an unwanted greenhouse gas to a sellable metal and oxygen by the following reaction: 212CO2 -> 24Mg + 2O2
Magnesium is also stable at atomic masses 25 and 26, so if you happen to have carbon-13 or carbon-14 in there, you are still good for the most part. There is the option to fuse the newly created magnesium with a third carbon atom to make Argon-36 which is also stable if the noble gas is more your style.
This concept also works for methane, only you release hydrogen gas instead of oxygen. Alternatively, you can react the methane to make anything from nitrogen to neon, using some of that bonded hydrogen.
If you have some control over what form a transmuted element takes, then transmuting lithium into diamonds might be an option steeped in profit and madness. With a roughly 90/10 split of lithium-7 to lithium-6, there is the slight problem of getting almost half of your diamond as carbon-14, but the other two carbon isotopes (12 and 13) are stable. An interesting record for the future archaeologists.
On something that is a bit more Mad Science then Mad Business, with a machine that can fuse anything, you are in a unique position to research transuranium elements. The issue again is likely neutrons, but if you can directly make something in quantity, then it can be easier studied and a potential use for it could be found. Then you can corner the market on its production as only your plebotium-powered device can make it efficiently enough.
Depending on the exact interactions with molecules and other more complicated things, your Mad Businessman could have a rather shady business of body disposal. Just ensure that the chunks of body can fit in the machine and transmute its [Elements] into [Another Element]. I suspect it would be hard to find the body if all of its carbon was transmuted into magnesium for example. They pay you for disposal, and you can sell the resultant products.
This does depend on being able to break down molecules with the transmuter device though, which it might not do.
A thought is that by direct transmutation, the Mad Businessman will have pure samples of any element they desire without going through a complex manufacturing process. The most profitable thing might not be to sell the pure element directly to a consumer, but to use the created element in a process to synthesize something else. Depending on how the machine is powered and its consumption, you could market your product as "green" or "environmentally friendly", and potentially charge a additional markup for it
Nobody needs to know you make this product with the tears shed by the laws of physics when you run the machine.