Do I need planning permission to build a garden climbing frame?

Generally no as they are not considered to be permanent structures and the pitched roof heights are below 4m. However you may need consult you local authority if:

You live in a conservation area.
The equipment will take over more than 50% of your garden.
You wish to build in your front garden or a field which cause a change of use.

It is obviously common sense also to consult with your neighbours also to avoid upsetting them.