Infinite $1 bills would be inherently less valuable just because who wants to carry that much cash around? Say, for instance, that you wanted to buy a $30,000 car with your new windfall. Would you rather show up with a stack of 300 $100 bills in your hand, or several wheelbarrows full of 30,000 $1 bills?
What are you on about? I use some of those unlimited bills to pay someone to put some more of those bills in a bank. I've still got unlimited $1s or $100s, I've just also got a large number in my bank account that I can use instead, and replenish when needed
At a certain point though you would just have an account with more money in it than you can ever spend and it wouldn't really matter what denomination the bills were except to the poor schmuck who agreed to deposit them for you.