Start with computationally tractable:
Suppose we have M dimensions with N values for each dimesion. Assume all values used.
So full entries are: $$M^N$$
Suppose we aggregate over 1 dimension we get $$M^{N-1}$$ values and we can do this for N different dimensions yielding:
$$N M^{N-1}$$
Next one we will N choose 2 (columns to aggregate over) times $$M^{N-2}$$
Repeating we see that total number of aggregates is:
Now let us assume that N = M, then:
And finally:
$$ Total \leq N^{N} \sum_{i=0…N-1} \frac{1}{2^{i}} \leq 2 N^{N} = 2 \textrm{x Number of original entries} $$
Seems likely that we can do worse than this in real world cases. Would be nice to get an upper bound ….