Which genus of plants or animals has the most species?

It's hard to find exact numbers.

For plants, the Genus Euphorbia consists of approximately 2160 species.
https://en.wikipedia.org/wiki/Euphorbia

This link says that possibly the genus Senecio has more,
though this reference states it's been revised to more like 1500 species lately:
https://en.wikipedia.org/wiki/Senecio

This link states the genus Agrilus (Jewel Beetle), with 2265 species is perhaps the largest in the animal kingdom, though this is 60 year old research
[From fond4beetles.com 2010 article no longer online]

Wikipedia puts the Agrilus species count at over 3000.
Here's a list of them:
https://en.wikipedia.org/wiki/Agrilus

Other documentations states there's only 15000 beetle specias out of 450 genera, so the 3K count seems suspicious to me, though I would think that beetle would be the right answer for the animal kingdom

That should say out of 450 genera in that specific beetle family Buprestidae. There's hundreds of thousands of beetle species.