Just a side note, of sorts:
This whole issue is a great reminder of the fact that scales can be thought of as "subsets" of all possible notes, and that any "subset," including a subset of a subset, seems to conjure up a certain "feeling" which the pianist can exploit. Looked at this way, it's all good, or at least it's all possible- and a pianist need simply be aware of the "feeling" they're getting when using a particular subset. Of course, feelings are subjective and relative and all that, but it's still worth strong consideration.
Example: you can paint a picture using practically all of the colors of the rainbow. You'll get a certain "effect" when you do this. This is kind of like soloing with lots of chromaticism- using every note in the chromatic scale, which will give a certain effect.
Or- you could paint the same scene using only primary colors: red, blue, and yellow. This will create a different effect, and is analogous to using, say, the blues scale (including the flatted fifth), which will give you a different, certain effect.
Yet again- you could paint the scene once more using only red and blue, and get yet another effect. This might be like playing with the minor pentatonic scale (with no flatted fifth), which will have its own sound, or effect.
I'm not trying to make an exact comparison- feel free to use your own colors (maybe, to you, using the pentatonic scale is like using "warm colors" or whatever). My point is this:
Limitation is a really cool thing! By using ONLY certain notes, you set up a certain "effect," which may be difficult to put into words but is there nonetheless. As an improviser you can explore different collections of notes and exploit the effect you get in each case. You could use the (6-note) blues scale, the minor pentatonic scale, or an even further subset, such as using only 1, b3, and 4. If you've never tried this, the effect can be really striking- it really does start to feel like a different sound, even if it may seem like simply using SOME of the notes that you USUALLY play. Likewise, you could create subsets of the mixolydian scale, such as only chord tones (1, 3, 5, b7) or the major pentatonic scale (1, 2, 3, 5, 6), and so on.
Of course, you CAN play anything, but that doesn't necessarily mean you SHOULD! Although the original question was "what can I play," (in which we could list several options, from a few notes all the way to the chromatic scale) I would agree with others that the 6-note blues scale is a good place to start. I tend to start most of my students with it (http://www.betterpiano.com/archives/how-to-play-the-blues-in-ten-minutes
) because I believe that this scale, with the flatted fifth included, is closely tied to the history and the heritage of blues music. When one uses it, one shows that he or she is grounded to this history, and has "paid their dues." In the end, though (and I realize that this is a more mature idea) an artist can choose what to use or exclude, taking into careful consideration the consequences of whatever decision(s) he or she makes.