crazyh

The school year starts Sept 11 in Greece for 2019 and usually the same around that time for most of Europe. By that time most vacations are over and families are gearing up for the school year, so I doubt there'll be lots of school age kids around.

If you can hold off until that time you should start seeing far less tourists, possibly more older tourists but still far less than the High Season. Naxos doesn't really ever shut down, especially Naxos Town where most businesses stay open. However, there'll also be shops, markets, and tavernas open for locals all over the island which may give you a more "Greek" experience.

As far as Rhodes, I haven't been there but from all accounts it's one of Greece's most popular and heavily touristed islands. If that's something you like it could be an alternative, but Naxos is highly recommended by many seasoned Greek Travelers.

That doesn't mean there aren't many other Greek Islands to go to that can offer what you want:

Amorgos
Tinos
Chios
Sifnos
Milos
Crete

to name just a few.

I think by early-mid September you'll find most islands pleasantly busy but without hordes of tourists, except for some of the Big Name Islands, even there crowds thin out until the cruise ships show up.

Heimdall

Travelerjan, I simply wanted to set the record straight so others reading this thread know how good Naxos is for teens. You knew that, and I knew that, but it didn't come out that way in the post.

Crazyh, you have never been anywhere in Greece in late August/early September so you have no way of knowing first hand. I have been there many times during that exact period, and know what goes on. Many families leave at the end of August, while others stay for the first week of September. And September 11th is the second week of September just as I said.