Apart from the Jerome Gambit (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+?? Kxf7 5.Nxe5+) and the Chicago Gambit (1.e4 e5 2.Nf3 Nc6 3.Nxe5??), the weakest opening that has a generally accepted name may be Damiano's Defense (1.e4 e5 2.Nf3 f6?). It was named for the Portuguese master Pedro Damiano (1480–1544). What a fish this Damiano must have been, you say, to advocate such a horrid opening!
Au contraire. Damiano rightly condemned 2...f6? as weak, and advocated 3.Nxe5!, which has for about 500 years been accepted as its refutation. George Walker, one of the most popular British writers on chess in the mid-19th century, accordingly wrote of 3.Nxe5!, "This constitutes the Damiano Gambit." George Walker, The Art of Chess-Play: A New Treatise on the Game of Chess (4th ed. 1846), p. 236. Later writers failed to draw the distinction between the move Damiano advocated (3.Nxe5!) and 2...f6?, so today poor Damiano is blamed for the whole defense.
Anyone who ever plays 1.e4 should learn the bust to the Damiano. After 3.Nxe5!, Black's best move is 3...Qe7!, which eventually led to a draw in Fischer-McGregor, simul 1964 after 4.Nf3 (4.Qh5+? g6 5.Nxg6 Qxe4+ 6.Be2 Qxg6 wins a piece) d5 5.d3 dxe4 6.dxe4 Qxe4+ 7.Be2 Bf5 8.Nd4 Nc6 9.Nxf5 Qxf5. However, as Dennis Monokroussos' analysis shows, Black is still in very bad shape in that line. Another possibility for White is 8.0-0 Qxc2 9.Qe1 Be7 10.Nc3 Nc6 11.Bb5 0-0-0 12.e3, threatening to trap Black's queen with 13.Ne1 - a line that, as Edward Winter observed in Chess Note No. 1934, the Weiner Schachzeitung advocated in 1912.
Instead, Black usually plays 3...fxe5?, allowing the obvious and strong 4.Qh5+. Then Black could try to trap White's queen with 4...g6 5.Qxe5+ Qe7 6.Qxh8 Nf6 (threatening 7...d6, 8...Kd7, and 9...Bg7). Unfortunately for Black, White gets to play moves too, e.g. 7.d3 intending 8.Bh6 or 8.Bg5. The main line is 4...Ke7 5.Qxe5+ Kf7 6.Bc4+, when 6...Kg6 would run into 7.Qf5+ Kh6 8.d4+ g5 9.h4 and wins. Instead, Black's best is 6...d5, allowing 7.Bxd5+ Kg6 8.h4! The key move, threatening 9.h5+ Kh6 10.d4+. Black must play 8...h6 or 8...h5, leading to similar play in either case. Then White, who already has material equality in addition to a raging attack, wins still more material with 10.Bxb7! (attacking Black's rook), when 10...Bxb7? falls into 11.Qf5#. Instead, Black's only real try is 10...Bd6, when White's best response is 11.Qa5!, preserving White's threats. Since 11...Bxb7? still gets mated, the only way Black can save his rook is the abject 11...Nc6 12.Bxc6 Rb8. Now White, who has just recovered the sacrificed piece, can gain a five-pawn advantage with the greedy 13.Qxa7 (not 13.d4?? or 13.d3??, Bb4+). Rybka instead recommends 13.e5 Bf8 14.Be4+ Kf7, assessing White's advantage as equivalent to 4.7 pawns. Unfortunately, in the only published game in this line (given below), White (rated 1677) played the somewhat inferior 13.Nc3 and managed to lose. Although White's position is a little loose, reasonably careful play should allow him to convert his enormous material advantage.