There are a handful of programs that allow you to cut costs while in Hamburg, although as always with such schemes you need to visit lots of places in a short time to make them worthwhile. Options include:
Hamburg Card Offers discounts on entry to museums, theatre tickets and harbour tours, and free public transport (including harbour ferries). You'll need to plan well and read what's covered carefully to make it worthwhile. Purchase online or at any tourist office.
Hamburg City Pass Includes entry to most Hamburg museums and covers free public transport, a free harbour tour and free bus sightseeing. You'll need to keep moving to make it worthwhile, but it's generally a good deal. The pass can be bought online.
Kunstmeile Hamburg Five of Hamburg's art museums offer a joint admission ticket that can provide great savings. Buy it at the museums. The standard version lasts for 12 months, but the cheaper three-day version is great value; the latter must be used over consecutive days.