Will BestBuy replace at no cost an iPod Touch bought 3 days ago thru BestBuy Express?

Prbably around half what you payed for the device, since it’s expensive, difficult and with the labour charges on that. Why should they replace what some random person bought from them. You broke it, You pay to fix it.

And how do you manage to break it in 3 days?

If you bought it with a credit card (which you likely did if you bought it at a kiosk) look into whether your credit card company offers full-replacement for the first 30 or 60 days on purchases. I know a few people who bought expensive gadgets (one friend an iPhone, another a laptop) and accidentally broken them and were able to get complete free-of-cost replacement of their gadgets because they’d purchased them with an American Express card.
Amex, Discover and a few of the other cards offer this “no-questions-asked-full-replacement” warranty of items purchased…even if it was because you broke it due to your own negligence.

If your card doesn’t offer this you’re best off to go to the Apple Store rather than Best Buy. Make an appt. at your local Genius Bar and hope they can replace it under warranty (free) or for a low cost. Best Buy will NOT be able to repair your Apple Device and you are entirely at the mercy of their in-store customer service which may or may not be good… You’re better off with the Apple Store folks.

Unless you bought a protection plan on the iPod, almost definitely not. (I work at Best Buy)
However, I would guess that the iPod does have a one year warranty on it. I would check to see what the warranty covers and take it into your local Apple store if it is covered.

However, there may be a way to get around the system at Best Buy if what happened to your screen isn’t covered in the warranty. First off, @geeky_mama is incorrect- Best Buy Geek Squad can and will fix the screen on your iPod, or they will replace it completely if they are unable to fix the iPod. However, my guess is that this would cost a significant amount of money if you don’t have a protection plan. If you still have your receipt, you can bring it in and get the cheapest, shortest plan they offer within 30 days of the purchase. You can bring your iPod in a couple of days later and they will fix or replace the iPod under the plan. It won’t be completely free of charge, but it might be more cost-effective than the other options if it’s not covered under your warranty. Good luck, and I hope this helped!