Now that the hard-science tag is gone, I will make a argument for life in several parts. I am not going into great detail, since this each topic is pretty broad in its own right, but I will attempt to demonstrate that there is a possibility for
There are chemical sources for organic materials on Jupiter
Jupiter's atmosphere is enriched in nitrogen, sulfur, and noble gasses by a factor of three compared to abundance in the sun. Since the noble gasses are not of any value in metabolic processes, the most abundant elements, in order, are Nitrogen, Carbon, Sulfur, and Phosphorous. There are concentrated cloud layers of water, ammonium hydrosulfide, and ammonia in the clouds as well. Ammonium hydrosulfide will form when there is sufficient ammonia and hydrogen sulfide and will enter in to chemical equilibrium with those latter two.
The carbon in carbon based lifeforms is advantageous because it readily forms chains and rings with complex geometry. It is the complexity of these geometries that allows so many different proteins that perform various life functions. In order to have even unicellular life, we need a chemical base that can also take many complex forms.
Carbon is present as a possibility for life forms. Sulfur is also present in about the same concentration as carbon, and has the advantage of being concentrated in the cloud layers. There is a distinct lack of oxygen, but there is a presence of phosphorous that can be used as an alternative in many cases. Sulfur has many unique possibilities as a base for life. One of its primary drawbacks, reactivity, is somewhat mitigated by the frigid temperatures present in Jupiter's clouds. It forms a variety of elemental allotropes, which gives it much of the flexibility of carbon-hydrogen chains regarding molecular geometry. Sulfoamides can create complex combinations with sulfur, carbon, and nitrogen.
The possibilities are extensive, but so I will just say that the concentration of ammonium hydrosulfide in the middle cloud layer, concentrates nitrogen and sulfur, which could potentially be the base of a life-system.
These sources can be replenished over time
Replenishment of organic compounds is crucial to the continuance of life. If lifeforms link together nitrogen and sulfur chains into particles large enough to sink into the planet, there needs to be some sort of cyclical mechanism to return the needed 'nutrients' to the clouds.
The evidence for this is simply that these elements, comparatively rare compared to the mass of Jupiter, are concentrated in the cloud layers, and have not been pulled by gravity into the center of the planet over the billions of years of Jupiter's existence.
There are sources of radiation energy on Jupiter
Jupiter's radiation belts differ from the Van Allen belts on earth in that they intersect with the atmosphere at the equator. These belts consist of solar radiation of charge particles trapped in Jupiter's magnetic field. Since these belts intersect the atmosphere, there must be a significant spray of radiated energy caused by collisions in the upper atmosphere around the equator.
An alternate radiation source is Jupiter's strong radio generation. Jupiter emits synchorotron radiation in the GHz range from its radiation belts, and decametric radiation from movement of charged particle between Jupiter's poles and its magnetodisk. Both of these radiation sources would emit in all directions, so a significant amount of it will be directed inwards to the planets surface.
Neither radiation source is as powerful as the sun's rays are to Earth, though I can't find precise enough measurements to estimate energy available. But these energy sources are there, as is the (much weakened) solar radiation.
The principles by which photosynthesis works can be easily applied to any of the other wavelengths present, as shown by radiotrophic fungi. Jupiter presents a case where a variety of autotrophs, each specializing in their own spectrum band, would be advantageous.
There is sufficient turbulence to keep 'plankton' in Jupiter's upper atmosphere
I asked a question related to this on Astronomy.SE. Basically, the ammonia, ammonium hydrosulfide, and water clouds that exist on Jupiter are all about an order of magnitude denser than the hydrogen/helium atmosphere, yet they stay aloft due to turbulence.
While the exact mechanics of Jupiter's cloud systems are not well understood, the evidence of the clouds themselves in significant. If water droplets can remain suspended in the atmosphere, than life forms which are less dense than water can remain suspended as well.
If an autotrophic food source is possible, then higher life is possible
In short, an obvious next step from plankton would be a jelly-fish like organism. With a small gas bubble filled with hydrogen, a Jovian cloud-jelly could maintain buoyancy in the clouds. Adding dangling tentacles or fine fan-like structure to filter out plankton would be a reasonable form of life.
In general, the presence of strong winds would give life-forms a variety of ways to keep aloft even if they increased their density above that of the hydrogen-helium atmopshere. There would be many possible sponge, polyp, and jelly-like organisms possible to form a complete ecosystem in Jupiter's atmosphere.