I found a game on Tim Krabbe’s site Journal Entry #278) that gives a unique example of 6 pawns on the same file, 3 for each side! Here is the game. The 6 pawns in a row occurs on move 21.
This is the record, and the limit, for most pawns on the same file by both sides in a professional chess game.
[Title "Sevcikova - Blichova, ch girls u12 Slovakia, 1999, 1-0"]
1. e4 e5 2. Nf3 Nf6 3. Nc3 Nc6 4. Bb5 Bb4 5. a3 Bxc3 6. dxc3 Nxe4 7. Bxc6 dxc6 8. Qxd8+ Kxd8 9. Nxe5 Rf8 10. O-O Bf5 11. Be3 Nd6 12. Rac1 Re8 13. Rfe1 f6 14. Nd3 Nc4 15. Rcd1 Ke7 16. Bxa7+ Kf7 17. Bc5 Rxe1+ 18. Rxe1 Re8 19. Rxe8 Kxe8 20. b3 b6 21. bxc4 bxc5 22. a4 Bxd3 23. cxd3 Kd7 24. a5 Kc8 25. f4 Kb7 26. g4 Ka6 27. Kf2 g5 28. fxg5 fxg5 29. Kf3 Kxa5 30. d4 Kb6 31. d5 cxd5 32. cxd5 c6 33. d6 Kb7 34. Ke4 Kc8 35. Kf5 Kd7 36. Kxg5 Kxd6 37. Kh6 Kd5 38. Kxh7 Kc4 39. g5 Kxc3 40. g6 c4 41. g7 Kb2 42. g8=Q c3 43. Qb8+ Kc2 44. Kg6 Kd2 45. Qd6+ Kc2 46. Qxc6 Kd2 47. Qd5+ Kc2 48. h4 Kb2 49. Qb5+ Ka1 50. h5 c2 51. Qc5 Kb1 52. h6 c1=Q 53. Qxc1+ 1-0
The record for the most by one side is four, and it is shared by many. No case of quintupled pawns has occured yet. But the most notable example is a game in which the structure survives an astonishing 23 moves.
[Title "Gabor M Kovacs-Rainer Barth, Balatonbereny Open, 1994, 1/2-1/2"]
1.e4 Nf6 2.Nc3 d5 3.exd5 Nxd5 4.Bc4 c6 5.d4 g6 6.Nge2 Be6 7.Bb3 Nxc3 8.bxc3 Bxb3 9.axb3 Bg7 10.O-O O-O 11.f4 Na6 12.Ba3 Re8 13.Qd3 Qb6 14.f5 c5 15.fxg6 fxg6 16.Qc4+ e6 17.dxc5 Qc6 18.Rad1 b5 19.Nd4 Qxg2+ 20.Kxg2 bxc4 21.Nb5 Reb8 22.bxc4 Rc8 23.Nd6 Rc6 24.Ne4 Rac8 25.Rd7 R6c7 26.Rd6 Rc6 27.Rfd1 Bf8 28.Rxc6 Rxc6 29.Rd8 Kf7 30.Rd7+ Be7 31.Rxa7 h6 32.Bc1 g5 33.h4 gxh4 34.Bf4 e5 35.Bxe5 Re6 36.Nd6+ Kg6 37.Bd4 Nb8 38.Ra8 Nc6 39.Rg8+ Kh5 40.Nf5 Rg6+ 41.Rxg6 Kxg6 42.Nxe7+ Nxe7 43.Kh3 Nc6 44.Kxh4 1/2-1/2
Interestingly, the 6 pawns game is an example for 2+1, 2+2, and 2+3 pawn structures. I wondered if 3+1 had ever occurred, and I found such an example. It occurs on move 10, which is probably the record for the fastest 3+1.
[Title "Jorn Sloth-Grigory Sanakoev, CORR, 1992"]
1. c4 Nf6 2. Nc3 e5 3. Nf3 Nc6 4. e3 Bb4 5. Qc2 O-O 6. Nd5 Re87. Qf5 d6 8. Nxf6+ gxf6 9. Qh5 e4 10. a3 exf3 11. axb4 Nxb4 12. Ra4 Re5 13. Qh6 fxg2 14. Rg1 Bg4 15. Rxb4 gxf1=Q+ 16. Kxf1 Rg5 17. Rg3 Rg6 18. Qf4 Qd7 19. f3 c5 20. Rb5 Bf5 21. e4 Be6 22. d3 f5 23. b4 b6 24. Bb2 f6 25. Kg1 Rf8 26. Ba1 Kf7 27. Bc3 Kg8 28. Qh6 fxe4 29. Rxg6+ hxg6 30. Qxg6+ Qg7 31. Qxg7+ Kxg7 32. bxc5 dxc5 33. fxe4 Rd8 34. Rb3 Kg6 35. Bb2 f5 36. Kf2 fxe4 37. Ke3 exd3 38. Rxd3 Re8 39. Kf4 Bxc4 40. Rg3+ Kf7 41. Rg7+ Ke6 42. Rxa7 Kd5 43. Rd7+ Kc6 44. Rd2 b5 45. h4 Bf7 46. Be5 b4 47. Rd6+ Kb5 48. Rd7 Bg6 49. Rb7+ Kc4 50. Rb6 Bh5 51. Rh6 Be2 52. h5 Rf8+ 53. Kg5 Bd3
This leaves 4+1 and 4+2 pawn structures still yet to known to me. 4+1 has probably occurred, but I severely doubt the existence of a 4+2. Additionally, based on the answers to this question, a case of queenside quadrupled pawns is also yet to be found.
Also, a game, if ever having occurred, with 4 pawns on one file for one side, and 3 on another for the other side, is not known to me. It would be very interesting if it has happened.
As an interesting side note, here is a shortish game in which tripled pawns occur on move 7. However, the actual record may be 6 moves or possibly even 5. Thanks for the game @Evargalo!
[Title "Gerhard Kiefer-Peter Krauseneck, Bundesliga Germnay, 1982"]
1.e4 e6 2.d4 d5 3.Nc3 Bb4 4.e5 c5 5.dxc5 Qc7 6.Nf3 Bxc3+ 7.bxc3 Qxc5 8.Bd3 Nc6 9.O-O Nge7 10.Rb1 Qxc3 11.Rb5 Nd4 12.Nxd4 Qxd4 13.Rxb7 Qxe5 14.Bb2 Qd6 15.Rb3 f6 16.Qh5+ Kf8 17.c4 dxc4 18.Bxc4 Qc6 19.Bb5