Under what circumstances would a foreigner require a re-entry permit? – and for how long?

A foreigner with a valid residence permit, who wishes to travel out of Denmark and to subsequently return to this country at a point when the residence permit is still valid, does not require a re-entry permit.

Any foreigner who is making an application for residence for the first time is able to obtain a re-entry permit, if there are reasonable humanitarian grounds for travel, or if more than three months have elapsed since the application was made.
 
Any foreigner who has made an application for an extension of their residence permit can be issued with a re-entry permit unless there are circumstances to suggest that the residence permit will not be extended.

Re-entry permits are time limited according to specific criteria.

Children of foreigners with permanent residency in Denmark are able to get proof of a residence from the Danish Immigration Service / police. This proof can be used as a re-entry permit. Proof of residency is valid for 3 years, but will however expire should one or both of the parents’ residence permits expire.