So the bill is $57 and from a $100 bill I receive back two $20s and three ones. I say the proper tip is $3. Your opinion?

Sorry i'm a little off topic -- I had to think about it for a second but is the "proper" change a twenty, two tens and three ones? It would make sense -- just want to know to see if I figured it out or if this is way over my head. Thanks.

American waiter, I presume, since you said dollars? Maybe he knew he gave you lousy service and didn't want to insinuate that you owed him anything more than $3.

Assuming you had no other [legal tender] banknotes, the proper tip was not $3 (assuming otherwise decent service). One of the $10 bills from the twenty you asked the server to break would be more like it.
I'm mildly horrified at foregoing the 67 airline miles by paying cash though.

I had somewhat of the opposite experience when I landed in Puerto Rico after traveling overseas for a year. At that point I knew I was back in the US by the way my change came back. I paid for a $12 meal with a $100 and got (roughly) 13-$1, 3-$5, 2-$10, 2-$20. I thought that was a bit overboard on the trying to get a big tip.

Probably more likely due to the waiter not having adequate change in their pocket. Hundreds are hard for waitstaff to break in cheaper restaurants.

$3.00 is a cheap-o tip.

When I'm traveling, I make sure I keep a stash of small bills separate from my other money just for tip purposes.

Me too.

Proper tip = $3?? People have lots of ways to calculate the value of service. Looking at your method from another perspective -- you gave the server about 5%. So are you being clever or cheap? How well did the server take care of you during the meal? Granted, at the very end of the service, the server did a possibly boneheaded thing and gave you change that couldn't be used to make a tip in any common amount. However (1) there could be other reasons besides being a bonehead -- what if he's out of smaller bills, the bar can't help, and he can't find a manager, so he decides to get you the check because he's concerned you've already waited a while while he runs around looking for change? (2) that last interaction was only a small part of the service received.

If you'd like change in other denominations in order to give a tip in a certain amount, would you ever consider asking?

Most people would indeed ask for change. But I think that the OP's point is a valid one. IMHO , part of professional, gracious service is for the server to go to the trouble of securing proper demoninations.

My very rough guidline for a moderate cost restaurant is to tip 20% of the pre-tax amount for good service, or 25% of the pre-tax for really excellent service. Once I had my proper change in hand, I'd probably reduce the tip, maybe by 5 to 10 percent of what I would have given otherwise.

The handling of payment is the final part of what should be an overall pleasant experience. To have the customer walk away from the restaurant on a negative note is not something I'd want, were I the restaurant owner or manager.

I don't think this was the case that time. I was at a mid price restaurant, eating a cheap meal at a fairly busy time. Change came from a central till not the individual waiters pocket.

Well, if you're not going back to the restaurant ever again, eh, just stiff 'em ...just kidding.
Just ask to break the twenty. Yes, it's kind of stupid to bring change like that but if the service was decent it would be pretty mean spirited to just tip three dollars for which there could easily have been a number of reasonable explanations for the denominations. On the otherhand, if this just topped off a really poor dining experience overall, I'd say sure, consider tipping the three.

Waitperson??? Snort, snort...

How about Waitunit? Waitbeing? Waithuman? Waitanoid? Waitominid?

Seems to me the waitperson made correct change, just not the way you wanted it

Unless your waiter did an absymal job, I am not sure why you think a 5% tip is appropriate. Clearly the waiter wasn't too bright to give you change like that so you just ask him to break the 20s in the denominations you want. Then tip based on his level of service to you.