What are the options for extending my visa as a USA citizen living in Phuket, Thailand without a retirement visa?

There is no multiple entry 3 month ANYTHING :O

the fact you rented a place here has no bearing on you getting a visa.

I take it when you flew in you got a 30 day visa exempt entry <- that means you flew in here without having a valid visa in your passport and were stamped in for FREE

You can extend that for 30days at the thai immigration offices in Phuket for 1900baht.

OR

you can go to a thai consulate and BUY a tourist visa <- it will be a single entry and get you stamped in ONCE for 60 days <- which you can extend for 30 days at the immigration office for 1900baht

OR

you can go to the border, stamp out of thailand, stamp in and out of another country and stamp right back into thailand to get a NEW 30 day vsia exempt entry stamp <- which you can extend for 30 days at the immigration office for 1900baht.

You only get Extension of Stay at Immigration Office in Thailand. At a Thai Consulate you can apply for a new Visa. As there is no multiple entry 3 months visa I guess do some research and let us know what kind of visa you are looking for.