The number of additional storable (permanent) alternate forms is
81 (27 Unown, 3 Deoxys, 2 Burmy, 2 Wormadam, 1 Shellos, 1 Gastrodon, 1 Basculin, 3 Deerling, 3 Sawsbuck, 1 Keldeo, 19 Vivillon, 4 Flabébé, 4 Floette, 4 Florges, 3 Pumpkaboo, 3 Gourgeist).
There are
33 additional Item-activated alternate forms (5 Rotom, 1 Giratina, 1 Shaymin, 17 Arceus, 1 Tornadus, 1 Thundurus, 1 Landorus, 2 Kyurem, 4 Genesect).
Out of these, Rotom, Tornadus, Thundurus, and Landorus are the only ones that can be stored in PokeBank in their alternate forms. Black and White Kyurem can be stored in the PC, but cannot be transferred.
You can transform Pokémon into
7 additional Battle-based forms (3 Castform, 1 Cherrim, 1 Darmanitan, 1 Meloetta, 1 Aegislash).
Battle forms cannot be boxed, so I'm not counting them towards the living dex.
Then there are 10 additional special forms (9 Furfrou, 1 Xerneas).
These cannot be boxed. Furfrou loses its haircuts when boxed, Xerneas only enters Active form during battle.