Depending on what theory, I've seen 10, 11, 17, and 21 dimensions. Kaluza-Klein theory began with 5. Kaluza discovered to his vast amazement that the Maxwell equations "fell out" of pure geometry if he assumed 5-D spacetime.
Most cosmologists now seem to believe (I'm not certain about the 'most') that there must be extra dimensions since string/M-brane theory seems to require them. The reason you can't 'go in that direction' is that all these extra dimensions are rolled up too tightly (~planck length) that as soon as you start out, you're back where you started! Hmm. Seems pretty weird, but the Universe just keeps getting weirder.
It would be cool to name the extra dimensions. "Widdershins" comes to mind....(?)
--Boris