You can artificially create as many dimensions as you like. For example, something on a 2-dimensional graph has dimensions which can be expressed by the coordinates (x,y). Something in three dimensional space can be identified with coordinates (x,y,z). If the object only has those coordinates at a specific moment in time, then we need to refine the coordinates to (x,y,z, t). However, several objects could be at that position, at that time, but each one could be emitting radiation of a different frequency, so we need coordinates such as (x, y, z, t, f). But there might be several objects at (x, y, z, t, f), with each one having a different Kelvin temperature, so we now need (x,y,z, t, f, k).
This can be continued ad infinitum.
Physicists might argue that the '5th and 6th dimensions', which I've suggested, don't meet the true criteria for 'dimensions' but mathematicians still make use of these concepts within modelling techniques, both for 'real-life' physical situations and to aid the theoretical physicists.
At university (when I studied mathematics) I taught myself to think in terms of 'n-dimensional space' where n didn't even have to be a rational number (e.g. pi-dimensional space) but alas, with the passing years, I'm now happier to stick with 3 or 4 dimensions :-)
Chris