As noted in the other post, there is no maximum thickness. A few years ago, an internal post office newsletter erroneously stated that the flat-rate envelopes had to be "flat," and despite USPS efforts to correct this, there are still some employees who insist that flat-rate envelopes must be flat.
Here's a memo that USPS issued officially stating that the flat-rate boxes and envelopes can bulge as long as they can be folded along the original flaps, and also clarifying that tape can be used. IMHO some of the text in the Q&A is pretty hilarious, and not only because it states three separate times that "“Flat” refers to the price, not to the shape."
https://about.usps.com/postal-bulletin/2016/pb22435/html/updt_002.htm
