This question touches on a pet peeve of mine, so bear with me a moment.
What most people mean when they say dimensions is "Realms of Existence," or "Worlds." Places that can be explored and inhabited. Nobody I know of has counted all of them, and you can consider them to be infinite in number for all intents and purposes.
The truer and, in my opinion, more useful definition of dimensions is the mathematical use, which is to say a direction of travel which exists at a right angle to all other dimensions. We recognize three physical dimensions, and travel through a fourth dimension we cannot perceive as easily which we call Time. We haven't mapped higher dimensions well either. String Theorists believe that there are only ten, other people believe other numbers. Personally, I think these are also infinite.
Realms exist with varying numbers of dimensions -- some higher and some lower than our own. This is why the issue is a pet peeve of mine -- if one calls the realms dimensions, it muddies your ability to discuss them properly. It's like calling coffee sugar, leading to questions like "how much sugar do you like in your sugar?"