How many times can you reroll cookie dough?

Hi, welcome to the forums! Yes, re-rolling the dough, particularly if you dust with flour, can affect the texture of the dough quite a lot. Working any dough with gluten in it activates the gluten and makes it tougher (more likely to spring back); likewise adding flour while dusting will dry and toughen the dough. And yes, resting and/or chilling the dough can help relax it to a certain extent.

That all said, I roll at least a few times, keeping flour to a minimum and I am fine. In fact, I generally like the performance of my dough better on a second roll or third roll, because the slight amount of added flour makes the dough spread less and hold its shape better. 

Every dough is slightly different, but if you bake off and taste small pieces as roll, you'll get the best idea of how your dough responds to the added handling and flour.

Thank you for the helpful tips.

I was going to say what Julia said. I would only add that if you roll your dough between parchment paper or plastic wrap, it won't pick up the extra rolling flour--plus, theoretically, if you pop the scraps into the refrigerator after cutting your first batch of cookies, the gluten should relax a bit and make your subsequent cookies less tough/chewy. (I never go that crazy. I just roll and cut shapes twice using flour, then make the remaining scraps into the baker's treats (for me!) To me, the third roll (the ones I keep and eat myself) are noticeably denser and chewier, so I don't want to serve them, anyway.   )

[I'm not a Food Scientist, but for the sake of discussion, I believe the science behind it is that gluten is formed only when water hits your flour in the making of the dough. There is water in your butter. (The oil in the butter also helps to avoid the development of gluten by coating the flour so water cannot get into it. I know that sounds like a Catch 22, but it seems obvious that the gluten formation wins out--especially if you handle (knead) the dough a lot.) That combination of moisture and flour continues to develop the gluten strands in the dough the more you mix and handle it. Gluten is what makes baked things chewy and dense; you don't want it to develop a lot in cookies and pastries--only in breads. So, the less you handle your dough, the lighter and flakier your cookies will be. These are things I have read and observed--as I said, I'm not a Food Scientist.]

In a cookie class I learned that some doughs should be rolled only once.   For decorated cookies I roll all the batch and cut, the leftovers in a ziploc, so when the first rolled batch is ready, I rolled the "leftovers"   because I noticed that if you bake  on the same  sheet  cookies from different "rolled"  they bake differently.

We re-roll the same batch until it's gone.

I find that by rolling my dough between plastic wrap and freezing the sheets, I always have dough ready. I do re-roll my dough between plastic wrap, chill and keep cutting. By using the plastic wrap I use almost no flour in rolling and my cookies maintain their shape in the oven because they're going into the oven cold. Hope this helps!!

I divide my dough into thirds. Then for each third I roll, cut, put all scraps to the side. I do this for my dough. Then I take all my scraps and roll those together. I try to reroll as few times as possible.