I get that. I took over 3000 photos when I was in-country a few weeks ago (new high zoom lense.)
Don't go bankrupting yourself this early in the relationship though. No one should be expecting fancy trips to Enoshima at this stage. Keep it simple. Make it about spending time together instead of going to wondrous places - save that for later.
There is plenty to do that won't drain your wallet and can provide photo opportunities. Street photography is legitimate photography.
Other ideas:
Showa Kinen Park in Tachikawa. Big bloody park with a JGSDF base next door. There is also a museum near the entrance (closest to Tachikawa station) dedicated to Emperor Showa. I didn't explore too much of it when I was there last, but Google Maps shows how big and varied the park is, so I bet there is plenty to do and see.
Can't believe I forgot this one: Koishikawa Korakuen Garden, right by Iidabashi/Suidobashi stations. Lovely landscape garden and only like 300 yen to get in if I remember right. However I don't know if its worth visiting in winter (we last went in sakura season).
Tokyo Government Metropolitan Building - the viewing platform on the top floor is free to the public, and the view is brilliant as expected.
You went to Skytree, so why not try the original Tokyo Tower - built to be slightly taller than the Eiffel Tower. You could pay money to walk up it, but it's just as impressive from the ground. More interesting colours than Skytree I think.
New Year's is coming up - there is bound to be some local festival you can visit. Very lovely for couples, which is a cliche but it's a cliche for a reason.
Ask her to teach you about her camera - that's a bonding activity that easily take an hour if she's technically knowledgeable about them. (I'm not, on the other hand. I shoot everything on auto and have no clue what most of the buttons do.)