Ludwig II did not build Hohenschwangau. That was his father, Maximilian II. Ludwig II did grow up there, though he was born in Munich's Schloss Nymphenburg.
Ludwig II then proceeded to build Neuschwanstein (left) on the next hill over from his parents' castle (right), a little bigger and a bit higher up:
You can walk from one to the other, quite a nice daytrip if you happen to find yourself in Bavaria.
While Hohenschwangau wasn't built by him, Ludwig II did build quite a lot of castles. Besides the already mentioned ones, there's also Schloss Herrenchiemsee:
It's modelled after Versailles. The gardens are lovely.
There's also the more modest Königshaus am Schachen:
The "Turkish Room" still packs quite a punch.
He also planned castles that were never built, such as a Neuschwanstein-style one at the site of the ruined Burg Falkenstein, not far from Neuschwanstein and Hohenschwangau.
This was contemporary theatrical painter Christian Jank's first rendering of that one:
Architects were engaged, but by that point Ludwig II was already in pretty significant financial trouble. A tiny bit of the infrastructure was put up, but nothing more. You can find some of the design sketches in the museum at Herrenchiemsee.
Shortly before his passing, Ludwig II also had plans for a castle inspired by a palace in Beijing, likely to be built in Tirol, and one that was to be more Byzantine-styled, likely near Linderhof.
He also did some lush royal apartments in Landshut and Munich, as well as other new additions to existing buildings, though a lot of that was cosmetic and/or leftovers from his bigger projects.