As far as I can calculate, 6 is the limit for the most number of pawns on a single file possible. Is 7 possible at all, ignoring promotion? I only took into account using the edge of the board, not the center, so I may be wrong.

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    If you don't count pawns on promotion squares (last row), then obviously you can only have pawns on the second to seventh row, I.e. max six pawns on a file. – user1583209 Feb 25 '19 at 6:28
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    If you do allow pawns on promotion squares, you need 6+10=16 captures, which is impossible. – user1583209 Feb 25 '19 at 6:33
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    To get to the same file, with minimum number of captures you need pawns from three files on one side and four files on the other side. Those are 1+2+3 =6 captures from one side and 1+2+3+4=10 captures from the other side. – user1583209 Feb 25 '19 at 7:07
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    oops. Indeed you only need 6+6=12 captures (three pawns from the left, three from the right), because you already have one pawn on that file... – user1583209 Feb 25 '19 at 14:27

If you only include legal positions then 6 is the limit. The pawns could possibly occupy the 2nd-7th ranks, via diagonal captures towards the desired file.

If you ignore promotion and allow pawns to be on the 8th rank (or 1st rank for Black's POV) then the limit is 7.

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Using black pawns-my thanks goes to @DM for this idea,-I have completed my quest to find the shortest possbile septupled pawns proof game in 18.0 moves.*

[FEN ""]

1. d4 b6 2. d5 a5 3. d6 exd6 4. Qd5 c6 5. e4 cxd5 6. b4 f5 7. Bd3 fxe4 8. c4 exd3 9. c5 bxc5 10. Be3 axb4 11. Nc3 bxc3 12. Bd4 cxd4 13. O-O-O g5 14. f4 gxf4 15. Rd2 cxd2+ 16. Kb1 f3 17. Ne2 fxe2 18. Rd1 exd1

Addendum-It turns out that the shortest possbile games is actually in 16.5. moves.

[Title "Henrik Juel & James Malcom ,PDB Website 07/07/2020, Proof game in 16.5 moves"]
[FEN ""]

1. g4 Nc6 2. g5 f6 3. gxf6 Nd4 4. fxe7 Nf6 5. exd8=B Nd5 6. a4 b5 7. axb5 c6 8. bxc6 Bd6 9. b4 Ba6 10. e4 Bd3 11. cxd3 O-O 12. cxd7 Rc8 13. exd5 Rc5 14. bxc5 Rf3 15. cxd6 Re3+ 16. fxe3 Kh8 17. exd4
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