The University of Copenhagen's Botanical Garden sits next to the Natural History Museum of Denmark and backs up against Rosenborg Castle. It is a pleasantly old-fashioned scientific garden — more research institution than public park — with a large Victorian palm house at its centre, herb and rock gardens, and a collection of around 13,000 plant species.
Entry is free and the garden is open year-round, though the peak season is May to October. The palm house (Palmehuset) was built in 1874 and is a beautiful piece of Victorian ironwork, still housing an impressive collection of tropical plants. On a rainy day it is one of the warmest and most pleasant places in the city. Combine with a visit to the Statens Museum for Kunst next door.