I think this might be the problem (assuming we're only allowed to take the factorial of positive integers) discussed in Donald Knuth's "Representing Numbers Using Only One 4", originally published in Mathematics Magazine 37 (1964), 308-310, and republished with addendum in 2011's "Selected Papers on Fun and Games", chapter 10. (It looks like this was also cited in the 1967 printing of Dudeney's "536 Puzzles and Curious Problems", edited by Martin Gardner, page 249.) I haven't been able to find an official PDF of the original that isn't paywalled, but Knuth shows that all numbers less than 418 at least can represented by starting out with one 4 and then applying a sequence of factorial and floored square root operations, as follows (which I haven't verified!) - Construct a directed graph with vertices labeled 3...999, with an edge from vertex y to vertex x if there exists some positive integers u and v such that x = floor((y!)^(1/2^u)) or x = floor((floor((y!)^(1/2^v))!)^(1/2^u)). - Then this directed graph on 997 vertices contains a strongly connected component of size 963 that contains all vertices < 418. - In particular, this means that there's a path in this graph from 4 to any vertex < 418, and so any positive integer less than 418 (at least) can be expressed this way. I'm not sure what the state of the art on this problem is, but it looks like the table in A139004 in the OEIS might have an expression of this form for positive integers up to 1000! (See also Wei-Hwa Huang's "123456789 = 100", http://www.gathering4gardner.org/g4g11gift/Huang_Wei-Hwa-123456789_Equals_10... .) --Neil Bickford