Every time I get a visa extension, they always ask me to bring an address or a proof of address from my apartment or hotel I am staying at. Exactly what do they want me to bring and what can I tell the hotel manager in order to get it?

It depends upon the immigration office you apply for the extension at.

A receipt for the hotel and/or a business card for it would be enough at many offices.

When I did my 30 day extension in January, (CM) it was one of the things you had to fill in on that form.  Just address and phone number.  As I don't have a phone myself,  the IO just asked me for my room number.

At Chaeng Watthana, I've never had to present actual proof, but they do insist that the address you fill in is complete by Thai standards (many people seem to struggle with the district and sub district). If you happen to stay at a hotel, they also want your room number, otherwise your unit number if you stay at a condo or apartment. I've seen them make people make calls to fill in any gaps, so it's best to be prepared.

 

When I once applied at Nong Khai immigration, it was basically the same, but they checked if the hotel had filed an address report for me. So I'd guess that you probably need to show proof if they cannot find that.

At Jomtien, if it is a hotel rather than a private apartment, just about anything will do. A receipt for your stay will certainly suffice. If they need more, they will phone the hotel while you sit at the counter. I have only seen one guy sent away from the TM30 desk and that was because he got shirty.

As always, it's the maddening answer for all Immigration questions here in good ol' Thailand...depends on to which Immigration Office you're going. That said... 

 

In Jomtien 3-4 years ago, after years of not asking for anything, as he would always do an extension on his annual visit, they sprung on a buddy that he needed a computer printout from the hotel showing proof that he was a guest...or no extension. At that time anyway, it had to be a computer printout. No handwritten paper, letterhead or business card. He explained that he was staying at a small hotel which didn't even have a computer. For ฿1000, the photo shop (right there on the property) had the necessary printout (complete w/bogus proof of residence!) for him in 15 minutes! Typically Thailand, but problem solved.