The first obvious one is Sephora! You don't need to buy things from them, you just need to join their buyer reward program, fill out the info, and then once per year, during your birthday month, you can go into one of their stores, give the clerk the email address associated with your account, and pick up your free birthday gift (they usually have at least 2 different gift options, and it changes every year). You can also have it shipped to you, but that costs $6, unless you are buying at least $50 worth of their items anyway to get free shipping, or have their Flash Shipping option.
Another one that is fun is getting a Free Burger from Red Robin (if you have one near you) and that one is also as easy as being a member of their reward/frequent buyer program. Also, if you have kids, this is a great and affordable place to eat and for every 10 meals, you get a free one. Because we are a family of 7, I rack up the free food there pretty fast.
DSW (Designer Shoe Warehouse) gives you a gift code for $5 off $5 or more for your birthday (which also means being a member of their reward program). Granted, I'm sure they think you'll go buy something and just get it for $5 less... but I go in there and find something is exactly $5 or slightly more, like a pair of socks or an accessory, and then I'm done. Free cheap item! I guess you can also just use it to make a pair of shoes slightly cheaper, too! But, the point is that there is no minimum purchase on that $5 off code (sometimes their codes require you spend like $25 or more or something like that), so you can literally get something free if it's $5. So I got free Steve Madden stockings.
Another one I used recently is from Soma Intimates. I've been a member of their reward program for 5 years and I think that's also how long it's been since I bought anything from them. BUT, I received a $10 off $10 or more birthday code from them, which is also STACKABLE WITH THEIR SALES AND PROMOTIONS!!! Right now, they were having a clearance sale that knocked an extra 70% off some of their underwear if you bought 5 or more... combined with a sale code they have that takes 50% off your most expensive item if you buy 2 or more, and then my own $10 off birthday coupon... I got 5 pairs of underwear for free, and a bra for $8. If I didn't want the bra, my order would have been entirely free, but I re,ally wanted the bra. You can use the coupon in store or online, but online you'll pay $5.95 shipping, FYI. But free underwear!
Starbucks is on this list not because I think it's good coffee (I'm from Seattle, we have MUCH better options that don't taste burnt), but a lot of people like it, it's everywhere, and they'll give you a free drink or treat for your birthday and a 15% off coupon if you join their reward program. So, you don't have to drink their coffee that tastes like cigarette butts, you can get a free treat instead!
I know this list isn't every single place that gives you freebies for your birthday, but those are the ones that I personally use and enjoy, and none of these require that you actually be a paying customer. If there are any that you have used that you think deserve a mention, let me know in the comments!