Well. I thought I’d finally come up with a tricky enough challenge that it might go un-guessed. Silly me.
The Geek in Question answered correctly in just 20 minutes. The mystery creature that lays an egg 1/3 its own weight is an ant cricket, Myrmecophilus. So, 10 points to TGIQ.
These small crickets are commonly seen in ant nests where they subsist on the ants’ secretions.