To exaggerate, overemphasize, overstate. An idiom. The 'over' is added unnecessarily and its usage is virtually always synonymous with a mere 'exaggerate', rather than a different word, as 'exaggerate' is not a neutral usage verblike the word 'do'. There is no threshold for the word 'exaggerate', so it is not possible to overdo. To 'overexaggerate' is to 'overoverstate'.
Overexaggeration differs from mesoexaggeration (commonly known simply as "exaggeration") in that mesoexaggeration strays from the truth to balance out skepticism, while overexaggeration goes beyond this to a non-believable, completely ridiculous extreme.
(actual example)
Tim: Me and my brother Joe took out four guys at this UFC fight thing. I can bench press 225 pounds. I got a 25...wait a 26...wait a 27 on the ACT.
Me: Tim, let's have a chat about this thing called exaggeration. Or, in your case, overexaggeration.