Bitfinex lets you do that, but you have to make a certain minimum deposit to start your account. I think it's about one bitcoin minimum deposit.
You have to verify if you want to withdraw as USD but not if you only withdraw bitcoins. Converting to USD then back to bitcoins is OK without verifying.
Okay thanks, that's worth to look at.
However I've heard that there's been a lot of problems around Bitfinex lately, are there any other alternatives?
Yes there are people worried it could go bankrupt if it hasn't enough money to back all the tethers it created.
There must be other exchanges that offer USD storage without verification, but I don't know of any. There's a lot of new exchanges opened that I'm not familiar with.