Although it's easy to replace the battery in most cars, it's a bit more difficult in an Audi TT, because you have to first access the battery itself, then move some complicated-looking wiring from the terminals. Fortunately, it just looks difficult, and isn't too bad to do.
Pop the bonnet and remove the plastic cover on the battery box using the 3/8-inch ratchet, extension and sockets. There is one bolt in the front by the core support, tucked behind the headlight, and another just behind the box itself.
Push the dimples in the middle of the plastic battery box to pop the panel free from the battery. Lift it up and away from the battery and place it to the side.
Disconnect the negative terminal from the battery using an open-end wrench, and place it to the side. Repeat this process with the positive terminal as well. Unbolt the wiring relay box on top of the battery using an open-end wrench, then pull it off and place it to the side so you can lift the battery out of the engine bay.
Drop the replacement battery into the battery tray and reinstall the wiring relay box using an open-end wrench, then reinstall the battery terminals, installing the positive terminal first, then the negative terminal. Push the battery box back onto the battery, then bolt everything in place using the 3/8-inch ratchet, extension and sockets.