They're all correct, but they have different meanings.
I am getting fat.
I am getting fatter
These two sentences are in the present continuous. They're talking about what is happening now - it's your current situation. The first suggests that you weren't fat before, but you are now becoming fat. The second suggests that you were fat before and you are now becoming even fatter. It's all relative, of course!
I get fatter.
I get fat.
These are in the present simple. They're talking in more general terms about what sometimes, usually or always happens to you. For example, 'Whenever I stop going to the gym, I get fatter. And then, whenever I start exercising again, I lose the weight'.