This weak defense gives white a large advantage after 3.Nxe5, but even if he chooses simple development instead he still gets an advantage because black's K-side has been badly weakened bu 2...f6.
The opening is named after the Portuguese player/author Pedro Damiano (1480–1544), despite the fact that even he condemned it as weak.
Obviously you'll never see it in master play. The last great player to use it was Chigorin who played 3...Qe7 against Schiffers at St. Petersburg 1897...Chigorin lost his Queen in the opening. You'd think he would have resigned then, but Schiffers' subsequent play was quite weak. Chigorin later missed a forced mate and only escaped when Schiffers agreed to a draw in a winning position.
It is possible to play it and may be get away with it at lower levels where sometimes just about anything will work.
Interestingly, in a 1964 simultaneous the 1955 and 1956 Wyoming state champion, 1979 Texas Amateur Champion and several several time Texas Senior champion Robert McGregor played it against Bobby Fischer and managed to draw.
Obviously, Fischer didn't play his best game, but even so, if it caught Fischer (who played better in simuls than most players we'll ever run into in a tournament) it might be worth a try at least once in a while. The game was part of a nationwide tour Fischer made in 1964.
[Event "Simultaneous, Houston"]
[Site ""]
[Date "1964.03.28"]
[Round "?"]
[White "Robert Fischer"]
[Black "Robert McGregor"]
[Result "1/2-1/2"]
[ECO "C40"]
[Annotator "Stockfish 15"]
[PlyCount "52"]
[EventDate "1964.03.28"]
{Damiano's Defense} 1. e4 e5 2. Nf3 f6 {According to Horowitz a strong reply
is either 3.d4 or the simple and potent 3.Bc4} 3. Nxe5 {This is the most
forceful because taking the N with 3...fxe5? exposes black to a deadly attack.}
(3. d4 {is best met by the rather surprising} d5 4. exd5 Qxd5 5. Nc3 Bb4 6. Bd2
Bxc3 7. Bxc3 e4 8. Nd2 Ne7 9. Bc4 {with a significant advantage.}) (3. Bc4 d6
4. d4 Nc6 5. O-O {and white is better, but there is no immediate, forcing way
to bigger gains.}) 3... Qe7 {This is black's best move.} (3... fxe5 4. Qh5+ g6
5. Qxe5+ Ne7 6. Qxh8 Nec6 7. Bc4 Nb4 8. Qxh7 Nd3+ 9. cxd3 d6 10. Qf7# {Vachier
Lagrave,M (2434)-Benmessaoud,Y Cergy Pontoise 2004}) (3... g6 {is just awful.}
4. Nf3 Bg7 5. Nc3 c6 6. d4 d6 7. Bf4 {White has a decisive advantage. Velchev,
H (2298)-Djermic,I (2077) Bosnjaci CRO 2010}) 4. Nf3 d5 5. d3 {The less
effective 5.Be2 was played in Ivekovic,A (1888)-Hosticka,F (2185) Nachod 2009}
dxe4 6. dxe4 Qxe4+ 7. Be2 Bf5 (7... Nc6 {as played by Chigorin is better, but
it's still a long way from equalizing.} 8. O-O Bd7 9. Nc3 Qg6 10. Ne5 Nxe5 11.
Bh5 {White is winning, but on move 34 he agreed to a draw in a won position.
Emmanuel Schiffers-Mikhail Chigorin St. Petersburg 1897}) 8. Nd4 (8. Nc3 {
is stronger. After} Qxc2 9. Qxc2 Bxc2 10. Nd4 Bg6 11. Bf4 {White's huge lead
in development is plenty of compensation for the c-Pawn.}) 8... Nc6 9. Nxf5
Qxf5 10. O-O Bd6 11. Bg4 Qb5 12. Nc3 Qc4 {Somewhat better was 12...Qa5, not
that it really matters.} 13. Be2 {With this move Fischer throws away all of
his advantage.} (13. Re1+ Nge7 14. Be6 Qb4 15. Ne4 {White has a winning
position.}) 13... Qf7 14. Bb5 O-O-O {The obvious threat is ...Bxh2+ winning
the Q.} 15. Qg4+ f5 {Much the best!} (15... Kb8 16. Bxc6 bxc6 17. Be3 {
followed by Qa4 leaves black in dire straits.}) 16. Qh3 Nge7 17. Ne4 h6 (17...
Be5 {Things get crazy after this!} 18. Nd2 Qf6 19. Re1 Bxb2 20. Ne4 Qd4 21.
Bxb2 Qxb2 22. c4 Nd4 23. Rab1 Qxa2 24. Nc5 Nxb5 25. cxb5 Kb8 26. Ne6 {and the
game could go either way.}) 18. Nxd6+ Rxd6 19. Bf4 Rd4 {It was probably safer
to play 19...Rd8} 20. Be3 Rb4 21. Bxc6 (21. c4 {keeps a slight edge after} Rxb2
22. Bc5 Re2 23. Bxe7 Nxe7 24. Qd3 Re4 25. f3 Re5 26. Qd4 Qf6 27. Qxa7 {with
the advantage.}) 21... Nxc6 22. b3 {At this point all of white's advantage has
vanished.} Re4 23. Rfd1 Rd8 24. Rxd8+ Nxd8 25. Rd1 Qe6 26. g3 Rxe3 {Draw
agreed.} 1/2-1/2
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