Natural maximization problems in FPT
Here are a few:


Max Cut: Can one color the vertices of an input graph $G$ black and white so that at least $k$ edges go from black to white?
Max Sat: Is there an assignment that satisfies at least $k$
clauses?
Max Leaf: Does $G$ have a spanning tree with at least $k$
leaves?
Longest Path / Cycle: Does $G$ contain a simple path / cycle on at least $k$
vertices? 
3-Set Packing: Given a set family ${\cal F}$ of sets of
size $3$, are there $k$ pairwise disjoint sets in ${\cal F}$?