3 dimensional universe looks like the limit for me. I can't even imagine a fourth dimension, and there are physicists talking of 11 dimensional spacetime. There are even mathematical generalisations of various vector analysis topics (which I'm just starting to learn) that extend it to 'N' dimensions. And rather than calling them perpendicular basis vectors they're being called 'orthogonal'. I'm wishing to have an easy and digestive explanation of all of this.
Mathematicians invented it as an abstraction, by just adding extra components to vectors and working out the properties of these new vector spaces.
These theories later worked themselves into the world of physics. Just like a 2D being wouldn't know about the third dimension, they considered the possibility of 3D beings (humans) not knowing of the fourth dimension, and of extra dimensions. For string theory to hold, a minimum of 11 dimensions would have to exist.