After the post I made a few moments ago, I saw how to do the count.
The answer's 3,199,724, which apparently is the standard answer accepted in Google.
The method is pretty straightforward. First, find the number of "partial racks" (i. e. sets of zero to seven tiles) consisting of just A; there's one with no tiles, one with one tile, and so on up to seven tiles. (There are 9 As, so a rack full of them is possible.) Then find the number of "partial racks" containing A and B; there's one with no tiles (the empty rack), two with one tile (A and B), three with two tiles (AA, AB, BB), and three with each of three through seven tiles. (There are only two Bs.) Repeating this for each of the 27 tile types (26 letters plus the blank) gives the answer of 3199724, not far from my sampling-derived estimate of 3224068.