OK, just off the top of my head:
Dussmann on Friedrichstraße - a fab play area, nice and empty in the mornings (they open at 10).
Streichelzoo (various farm animals basically) in Görlitzer Park in Kreuzberg.
Botanischer Garten is beautiful, at every time of year. (Costs to get in, though).
Sea Life aquarium on Unter den Linden.
Lovely playgrounds in Prenzlauer Berg at Helmholtzplatz and Kollwitzplatz (latter has a huge sandpit and baby swings).
The Johanniter (?) in Kreuzberg run a Winterspielplatz (basically big space with soft play, lots of toys/trikes/cars, etc.)
St. George's second-hand English bookshop in Wörther Str. (Prenzlauer Berg) has toys and rocking horse etc.
There is a Kindercafé somewhere - never been, but I think it's called 'Klax'.
Do the playgroups have to be English-speaking? The Anglican church (St. George's in Charlottenburg) runs a Parent and Toddler Group from 10am on Mon mornings. There is a bookshop in Schöneberg called Purzelbuch which runs music and movement courses for young children (German-speaking).
There's a lot more stuff for summer.
I don't live in Berlin (any more/yet - we're almost certainly moving back next spring, though) - admylin and SSSandy are 2 MNers who do. I have a 17-month-old.
HTH!