Comment by pfdietz

The model I like is assuming the numbers you manipulate in O(1) time are bounded by some polynomial in the size of the input. Or, O(log n) bits. This seems reasonable, as just addressing a size n input uses addresses of O(log n) bits.

One can do some interesting things in this model, such as the van Emde Boas data structure for O(log log n) integer set queries, and tricks where O(log n) size subproblems are converted to lookup tables.