The Philosophenturm, short Philturm, is one of the (not really beautiful but) biggest buildings on the campus of the University Hamburg. In 1960/61, the Philturm was built by Paul Seitz for the studies of arts and humanities.
After a really affordable lunch in the canteen (Mensa) in the first floor (check menu and prices on this website), you could go to the top by elevator or by feet (uff!) and have a nice break and watching the beautiful city from above FOR FREE. Take a coffee with you and relax in between students running e.g. from room 267 to room 865.
Take the elevator to the highest floor (I think it’s 12) and take the steps for the last two ones. It’s a bit irritating that there are no signs for the stairway but at least you could find it out by yourself. You need to go through glass doors and corridors to find the last step. There you can sit on the steps, let your legs dangle.
You can see the main train station, the Binnenalster (inner Alster Lake), the inner city (see picture), on the other side you can see the campus, the TV tower, the St. Michaelis Church (short Michel), the port of Hamburg, everything!
I love it & it’s free.