Troglodytes troglodytes
Crex crex
Pyrrhocorax pyrrhocorax
Puffinus puffinus
Cinclus cinclus
Meles meles
Lutra lutra
Crangon crangon
Melolontha melolontha
Typhaeus typhoeus almost made it.
I wonder if there is a pattern that the more active the taxonomist, the more they repeat themselves. Linnaeus was probably turning out 50 new species per day and once he had come up with one apt name, he maybe couldn’t be bothered thinking of another.
Mephitis mephitis. Can’t you just smell skunk in that one?
Also Dives dives, the Melodious Blackbird.
Motmot is an example of a linguistic tautonym but not a taxonomic one.
As a plant nerd you are really disadvantaged here.
Salacca zalacca is perhaps the closest thing to a tautonym. 
Mahi-Mahi?
Carduelis carduelis
Chloris chloris
Vanellus vanellus
Pica pica
The one that seemed really obvious to me is Tyrannus tyrannus.
Himantopus himantopus - African/Eurasian Black-winged Stilt
Earlier today, I saw my resident Cardinalis cardinalis at the bird feeder. I’m surprised they haven’t been mentioned yet.
Not allowed in the plant world! But sometimes we get close, like with:
Thalictrum thalictroides, you know, the Thalictrum that’s thalictrum-like.
(The near-tautonym is the result of a transfer between genera.)
My favorites:
Pica pica pica - a Magpie (say it fast, it’s great!)
Rattus rattus - a Rat
Lemmus lemmus - a Lemming
and
Puffinus puffinus - not a Puffin!
Similar things also happen in the animal world - take this African fly in a monotypic genus Meromacroides meromacriformis (not yet on iNat) - meaning it’s like the New World genus Meromacrus: “no - it really is like that”.
Caretta caretta, the Loggerhead Sea Turtle
Lynx lynx 
Which can only be definitively IDed by it’s pelvic bones. Other features aren’t diagnostic, but the hips don’t lie.
I choose you!
Anhinga anhinga
(It’s an anhinga - common name also! Does that make it a trionym?)
Alces alces – the Moose.
