I know that Saxifraga sarmentosa (Strawberry Begonia - it's not a Begonia despite the common name) grows plantlets on runners. It also (as far as I've noticed) does well on benign neglect however to a very slightly lesser degree.

This is my Spider Plant. It's in a 10in plastic pot with like, a million babies. I grew it from just a little tiny plantlet started 3 years ago off of my mother's plant, which she's since killed. Yours looks so happy!
It looks a little deflated in this pic 'cause it was thirsty when it was taken. It puffs up substantially when not thirsty.
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Yours looks adorable! and very happy too! Those Strawberry "Begonias" are very cute too! might havta add it to my list haha. Thanks for sharing the pic!!

Strawberry Begonias are cute. They're very tolerant and if you're lucky they'll send up a flower stalk in the middle of the plant. I, however, have never had mine flower for me...so I dunno what it takes.