Max memory used by mathematica to generate a factorial.

nFacMaxMemData.nb

(* data = { x , MaxMemoryUsed[ByteCount[x!]] 11.0.1 for Linux x86 (64-bit) (September 21, 2016) *)
(* through x=10^10 *)

data = {{10, 56},
  {12, 800},
  {15, 800},
  {19, 800},
  {25, 800},
  {31, 776},
  {39, 800},
  {50, 800},
  {63, 800},
  {79, 800},
  {100, 264},
  {125, 800},
  {158, 800},
  {199, 800},
  {251, 800},
  {316, 776},
  {398, 880},
  {501, 1072},
  {630, 1392},
  {794, 1776},
  {1000, 2416},
  {1258, 21664},
  {1584, 27168},
  {1995, 38688},
  {2511, 47520},
  {3162, 59552},
  {3981, 77344},
  {5011, 92576},
  {6309, 119648},
  {7943, 151968},
  {10000, 183880},
  {12589, 230072},
  {15848, 291592},
  {19952, 361544},
  {25118, 449720},
  {31622, 591096},
  {39810, 745800},
  {50118, 942384},
  {63095, 1189336},
  {79432, 1493200},
  {100000, 1869648},
  {125892, 2358984},
  {158489, 2999456},
  {199526, 3808256},
  {251188, 4790144},
  {316227, 6045392},
  {398107, 7659440},
  {501187, 9711936},
  {630957, 12242328},
  {794328, 15484944},
  {1000000, 19505752},
  {1258925, 25170496},
  {1584893, 31490008},
  {1995262, 39894688},
  {2511886, 50173552},
  {3162277, 63451792},
  {3981071, 80382728},
  {5011872, 101746304},
  {6309573, 128590144},
  {7943282, 163330792},
  {10000000, 206477432},
  {12589254, 262882904},
  {15848931, 331942832},
  {19952623, 419226504},
  {25118864, 531066416},
  {31622776, 674988496},
  {39810717, 851461880},
  {50118723, 1081772264},
  {63095734, 1368389928},
  {79432823, 1729475512},
  {100000000, 2195109288},
  {125892541, 2783204984},
  {158489319, 3521806112}, 
  {199526231, 4454780144},
  {251188643, 5643883736},
  {316227766, 7154343000},
  {398107170, 9057776912},
  {501187233, 11479044752},
  {630957344, 14532842568},
  {794328234, 18398616112}, 
  {1000000000, 23311655104},
  {1258925411, 29519574792},
  {1584893192, 37404395816}, 
  {1995262314, 47369471056},
  {2511886431, 59956997392}, 
  {3162277660, 75963576760}, 
  {3981071705, 96180248328},
  {5011872336, 121830360536},
  {6309573444, 154293267576},
  {7943282347, 195399783840},
  {10000000000, 247457123816}
}