How many different smells can a human recognize?
As nobody is suggesting an answer I'll give my guess that I already discussed in the comments and give some extra details about ongoing debates on the subject.

We'll start with the sense of color sight in order to make easier. One can argue that we see in 3 dimensions because we use three cells which level of activation inform the brain of the color. I guess that the whole 3D space delimited by the maximum level of activation of each three receptors can be used. Therefore, it is nothing else than the number of receptor that determine the number of dimensions in which we can sense. Now it is assuming that each receptor can send signal of different strength information of the intensity of the light or smell.

According to wikipedia. Mammals have about thousands of genes that code for olfactory receptors. Moreover, the question of how these receptors work is still debated


  In mammals, each olfactory receptor neuron expresses only one functional odor receptor. Odor receptor nerve cells function like a key-lock system: If the airborne molecules of a certain chemical can fit into the lock, the nerve cell will respond. There are, at present, a number of competing theories regarding the mechanism of odor coding and perception. According to the shape theory, each receptor detects a feature of the odor molecule. Weak-shape theory, known as odotope theory, suggests that different receptors detect only small pieces of molecules, and these minimal inputs are combined to form a larger olfactory perception (similar to the way visual perception is built up of smaller, information-poor sensations, combined and refined to create a detailed overall perception).[citation needed] An alternative theory, the vibration theory proposed by Luca Turin, posits that odor receptors detect the frequencies of vibrations of odor molecules in the infrared range by electron tunnelling. However, the behavioral predictions of this theory have been called into question. There is no theory yet that explains olfactory perception completely.


Note: the two competing theories for color sights: Opponent process and Trichromacy

Hope this helps!

I think this question would also be a good fit for cognitive science beta.