I think the phrase down the rabbit hole is actually referring to Alice in Wonderland. But.
I would say the organism that tends to burrow the deepest into the Earth is humans. Average oil well depth appears to be around 5,964 feet (1818 meters), that’s pretty deep. The deepest hole we ever drilled is supposedly the Kola Superdeep Borehole dug by the Soviets, it was 40,230 feet (12.2km) deep.
Oh, that’s a fair point on both counts, I should have specified non-human organisms. Still, we’re apparently really good at digging deep holes, so that’s fun!