We always encounter Conformal maps in Physics, may be they are easier to study, but are there Non-Conformal transformations encountered in Physics anywhere?
if they are encountered, where are they prominent and are they a hurdle to further theorize or model that particular phenomenon?
Conformal maps tend to be the exception rather than the rule. In general, if a transformation $T:S\to S$ on some $n$-dimensional space $S$ is of interest in physics, it will not be conformal. (Indeed, there's generally no guarantee of a useful notion of angle in that space, but even if there is such a guarantee then $T$ is still not likely to be conformal.)
This doesn't mean that they're not worthwhile objects to study. They do of course come up with some regularity, and when they do there is a lot more that we can say about the theory just based on its conformality properties. It will also often be the case that many theories (like fluids which are compressible but not too much) will keep qualitative features of similar conformal theories (i.e. incompressible irrotational fluids).
It's probably fair to say that conformal theories are slightly over-represented in undergraduate courses. This is perfectly natural because they're generally easier to study than the more general theories which do not have the conformality properties, as those tend to be a lot more messy and therefore left for later courses. However, make no mistake: they're very much special cases.
