People would sell their US dollars for Euros or CAD or whichever countries aren't doing this printing the moment they got paid.
Places have printed money like crazy before, and it can get bizarre if you do it enough.
The biggest issue with this aside from creating an unpredictable currency is that everyone’s effective tax rate would be the same, so you would have to reinvent the welfare system to recreate progressive taxation levels.
Imagine two people each get paid $1000 USD and the available currency is doubling weekly.
The first person simply holds on to their money in cash. At the end of the week their money is worth half as much.
The second person buys Euro's the day their paid, and one week later they can exchange back to USD for $2000.
That person effectively paid 'zero' taxes.