how long does it take for a cat to become full grown
It depends on the breed. Many of the newer breeds (ragdolls, rag-amuffins), and many of the larger breeds do not reach maturity till they are 2 or three. Of course, that doesn't mean they won't go into heat or start to tom, so do fix them. :-) most cat cats are about 12 -18 months for full size and maturity.