According to superstring theory there are 10 or 11 dimensions. The reason for two different numbers is extremely too complicated for just one post so I recommend you read Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension by Michio Kaku. That book covers how pointlike particles are treated as vibrating strings and the ways they vibrate allowing for dimensions.
Here is a short YouTube video about 10 dimensions:
Here is a short YouTube video by the author (Michio Kaku) about 11 dimensions:
The existence of extra-dimensional space is a vague concept. I say 'vague' not because that it doesn't exist but because how the concept of extra dimension is proven by physicists.
It is somewhat similar to that of the existence of a black hole. We cannot really observe a black hole directly, but guess about it by studying the phenomenon taking place at the 'event horizon' (the place from which things can still escape the 'pull' of the black hole). The crucial difference is that we know for sure that the black hole exists because they hold consequences for the universe around it but with extra dimensions, the scope of their interaction with our space is not really observable.
The general rule is that if something can be conceptualized, the possibility of its existence exist. The highers dimensions can not only be conceptualized but they can also be mathematically and rigorously treated. This hold true for any number of dimensions and theoretically there is no limit.
Note: Please remember that the same paradox is also true for lower dimensions.