Can I extend my 30-day tourist entry stamp in Thailand for another 15 days?

Yes just go to immigration with like a week to go and get another 30 days it’s about 1800 bhat or 1900 I do it all the time

As you arrived on visa exempt entry you are indeed allowed ONE extension that will be for 30 days and cost 1900 baht a passport photo and a few documents.This can be done within 30 days of your stamp running out but some immigration offices will send you away and tell you to come back when you have around 7 days remaining, the 30 days get added to the end of your visa so you don’t lose days for doing it early etc.

After this extension if you wish to stay longer you would then have to leave and re enter the country.

Depending on your point of departure, if you do not have a confirmed travel reservation OUT of Thailand within 30 days you may have issues boarding a flight.

What is your nationality and point of departure?

As long as you enter thailand on passport that is eligible for visa-exempt entry, you can be sure to get 30 days tourism extension. 30 day will be added to your stamp, you wouldn’t lose any day when you apply for it earlier. if extension is less than 30 days, you can ask the immigration officer to correct it.