How soon can I apply for an extension after entering Thailand on a visa exempt stamp?

In Bangkok & Chiang Mai you can do it when you have 45 days or less left on your current "admitted until" stamp (meaning on a 30 day visa exempt you could show up at the immigration office right from the airport to get one if you wanted to

Some immigration offices are 30 days or less and some are 15 or less left on your current stamp.

Remember they ADD the 30 day extension to your current expiration date so you don't have to wait until the last day to go get it.