What are the visa requirements for a US citizen planning to stay in Chiang Mai for more than 90 days?

Being you want to stay longer than 90 days, your other option, and better for what you want to do, is get a multi-entry tourist visa. This is good for six months, and inside that six months you can enter Thailand as many times as you want, getting 60 days each time. Then you can extend each entry for 30 more days for bt1900 at immigration inside Thailand. If you make your last border bounce a few days before this six month multi-entry tourist visa expires, you can get almost 9 full months in Thailand on just this multi-entry tourist visa, leaving Thailand and returning only twice. At the end of that 9 months, then you can leave, and come back for another potential 75 days, 45 days visa exempt stamp entry, and 30 day extension.

You can get only one extension off a tourist visa at a local immigration office- so 60+30 as you mentioned. After that you can do a border bounce and enter visa exempt. That would give you 45 days (30 after April 1st). You can also extend that once.

If you arrive without a visa you'll be stamped in for 45 days visa exempt (after March 31st this goes back to 30 days visa exempt).

If you arrive with a tourist visa, you'll be stamped in for 60 days.

Both of these can be extended one time for 30 days at your local immigration office.

If you want more time after that you'd need to leave the country. You cannot extend a visa outside of the country, you can only apply for a brand new visa at an embassy.