I signed up for a CPR number, which is needed for practically everything here. It's a national registration number issued by the Danish government. In about a week I should receive the number by mail.
With it, I can:
get universal free healthcare
open a bank account
buy a SIM card
This also means that I won't be able to do any of these things until then. I understand its necessity to government-related functions like healthcare, but to buy a SIM card?
Edit: Okay, you can buy a SIM card w/o a CPR number, but only the prepaid kind. If you want a contact plan with the phone company, which is cheaper, a CPR number is needed.