It's a Pelargonium of some sort. They are often commonly called "Geraniums", though they are not.

The foliage looks like that of a trailing Pelargonium peltatum hybrid. And probably a more moden hybrid with the burgundy color.

Outdoor, full sun, lots of fertilizer

Thanks! I should have figured some kind of "geranium," but it's so itty-bitty. I think I'll put it in a pot since it's so tiny and bring it in for the winter.

The burgundy color is really pretty.

It is only 4 inches tall at the moment but repotted in a big enough container it can make a much bigger plant. And it will trail so needs to be up high. It won't work in a flower bed. They are typically used in window boxes and containers. Yours is not a special type- it's just stunted. Look up how to get Pelargoniums through the winter. There are techniques.

Actually, under the right circumstances it will work planted in ground if you have a bed where it can be the only, or main, species. I use mass plantings of them in my terrace beds some years. They tumble over one another, and when they reach the edges of the terrace proceed to cascade down. Their little 'heads' of flowers raise above the foliage well and it's quite effective. Peltatums, though considered a full sun plant, do a bit better if they're not put in direct, blazing sun in the U.S. They like it much less than the hortorums and thrive in bright, filtered sun in many latitudes.

'....under the right circumstances...' Clearly you have more experience of this than I do, Calliope. I've only ever seen them in containers or raised planters where their trailing habit can be appreciated. But then I've only ever gardened in a temperate climate where they'd be lost in a flower bed at ground level. Although you do mention that they 'cascade' when they get the chance. I have them in my window boxes right now and they are just starting to flower here but not trailing much yet.

Looks like ivy geranium. I overwintered one on my windowsill but it was an aphid magnet.