Is "over-exaggerated" correct English?

Over-exaggerate is certainly in current use. The OED has three citations from 1900, 1928 and 1984 supporting the sense of the act of exaggeration which is in or to excess, too much, too.

Common Errors in English Usage covers this topic:

“Over-exaggerated” is a redundancy. If something is exaggerated, it’s already overstressed.

My initial thought is that over-exaggerated implies not only exaggerating, but exaggerating in a way that is excessive for the given context, or exaggerating to the point of absurdity. So, saying something like

The fish was 5 feet long!

I would consider exaggerating, but something like

the fish was a million feet long!

would be over-exaggerating.

I would say "exaggerated" is enough. However, "over-exaggerated" could be used for the sake of a play on words.

It's correct. It means to exaggerate in a way which is inappropriate(ly excessive) for the circumstances. Exaggeration is a technique used to create interest in a story, but it's over-use (obviously subjective - some would call x over-exaggeration, some wouldn't) is over-exaggeration.