Now to the analysis...
1. e4 Nf6 2. e5 Nd5 3. d4 d6 4. c4 Nb6 5. f4
The Four Pawns Attack of the Alekhine Defense, which is believed to give White an advantage. But as Black's first four moves seem quite sane, perhaps there is a hidden resource for undermining the White pawn center before it can be consolidated.
5...g5
This move has not done well in practice, but I believe it is viable if followed up correctly. Its purpose is to undermine White's pawn chain by pressuring its base at f4, while enabling Black's bishop to develop effectively at g7. Though 5...g5 weakens the h6 square and the b1/h7 diagonal, White's pawn advances seem at least as compromising, having created holes at d3, d4, e3 and e4. And as will be seen, the published refutations are readily sidestepped by Black.
Since many of the positions arising from 5...g5 are too complex to permit the enumeration of all plausible alternatives, I will focus on what appear to be the most critical lines.
Sharp tries by White:
A. 6. fxg5
B. 6. exd6
C. 6. d5
A. 6. fxg5
6...dxe5 7. dxe5 Qxd1+ 8. Kxd1 Nc6 9. Bf4 Bg7 10. Nf3 Bg4, and Black regains the pawn while remaining ahead in development.
B. 6. exd6
6...cxd6
Theory only gives 6...gxf4 7. dxc7 Qxc7 8. Nc3 which is clearly better for White, given the weakness of the f4-pawn.
7. fxg5
The pawn sacrifice has exposed White's d-pawn to pressure along the a1/h8 diagonal, accentuated by the tempo White has just expended. Moreover, the extra pawn's being doubled means that White's kingside pawn majority does not constitute a potential passed pawn.
7...Bg7 8. Nc3 Nc6 9. Be3 d5 10. c5
10...e5
After 10...Nc4 11. Bxc4 dxc4 12. Nge2, Black seems to have inadequate compensation for the pawn.
11. dxe5 d4 12. cxb6 O-O
Regaining the piece immediately would have permitted the disruptive 13. Qxd8+.
After 12...O-O, I have not found a sure path to a White advantage, but to call this position "rich in possibilities" would be a gratuitous understatement.
C. 6. d5
Here theory gives 6...gxf4 7. e6, but Black has a more appropriate response.
6...dxe5
Since White is bound to generate kingside threats, Black must open the center as a step toward counterplay.
7. fxe5 Bg7 8. e6
Now 8...fxe6 would be suicidal, for example, 9. Qh5+ Kf8 10. Nh3 h6 11. Bd3.
8...O-O
Black will need to rely on his lead in development to compensate for his fragile kingside.
To be examined:
C1. 9. exf7+
C2. 9. Nc3
C3. 9. h4
C4. 9. Bd3
C5. 9. Qh5
C1. 9. exf7+
9...Rxf7 10. Nf3 h6 11. Bd3 c6 12. dxc6 Nxc6 13. O-O Be6 14. Qe2 Nb4
15. Qxe6
15. Bg6 Rf6.
15...Qxd3 16. Qxf7+ Kxf7 17. Ne5+ Kg8 18. Nxd3 Nxd3 19. Nc3 Bd4+ 20. Kh1 Nf2+, with perpetual check.
C2. 9. Nc3
9...fxe6 10. Bxg5 Qd6
The threat of 11...Qe5+ gains control of the c5 square without loss of tempo. Instead, 10...exd5 11. c5 would be unpleasant for Black.
11. Nf3 exd5 12. cxd5 Bg4, and Black's pieces are very active.
C3. 9. h4
9...Qd6 (threatening ...Qg3+), and White's king seems the more vulnerable after 10. Rh3 fxe6 or 10. Qd3 g4.
C4. 9. Bd3
9...h6 10. Qh5 Bxe6 11. Bxg5
11. dxe6 Qxd3.
11...Nxd5
11...Bxb2 12. Qxh6 f5 13. Qg6+ assures White at least a draw, as 13...Bg7 14. Bh6 Rf7 15. dxe6 is crushing.
12. Bxh6
12. cxd5 Qxd5, and Black regains the piece with interest.
12...Nf4
"Simplest."
13. Bh7+
Gaining time over 13. Bxf4 Qxd3.
13...Kxh7 14. Bxf4+ Kg8, and White's in trouble.
C5. 9. Qh5
9...h6 10. h4
10. Bd3 Bxe6 transposes into Variation C4.
10...Qd6 11. hxg5 Qg3+ 12. Kd1
12. Kd2 loses a bishop to 12...Qf4+.
12...fxe6 13. Nf3 exd5 14. Rh4
14. Qg6 Bf5 15. Qh5 Bg4.
14...dxc4
Opens the d-file while preventing 15. Bd3.
15. gxh6 Bf6
White's attack seems to have run out of steam.
Conclusion: The move 5...g5 appears to hold up.