How many, how deep?

The average move number for both players to complete castling is move #20 (to the nearest 5 moves), so what is considered an opening probably extends only 2-3 moves past that--time enough to place the two rooks after castling.

Many good players recommend memorizing entire lines, not just a few starting moves, which would mean out to about 20 moves. I also recommend the same, except for uncommon openings (Bird's Opening, Bishop's Opening, Vienna Game, etc.) where typically only about 3 moves will suffice, *except* for highly tactical openings like certain gambits (King's Gambit, Latvian Counter-gambit, Elephant Gambit, etc.), where every move is critical for the first several moves.

Many good players recommend sticking to a single opening, if that's what you're asking. In other words, your responses should always be fixed, but the possible responses by your opponent increase so rapidly that you'll more than have your hands full even with that limitation. I agree with that advice, at least as long as you continue to like the fixed response you chose.

Here's my recommended memorization list to be learned for only openings you can encounter with your fixed choice of moves (therefore don't memorize this entire list):

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Sqod's recommended bare minimum opening moves memorization for club/novice levels.

These recommended move sequences do the following:

o contain only essential moves to know.
o avoid the most common early opening traps.
o develop the best knight first, in order to avoid early material loss or poor position.
o exploit the most common mistakes.
o leave off at a move where most natural continations will suffice.
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King's Pawn Opening (1. e4)
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Double King's Pawn Game (1. e4 e5)
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Bishop's Opening
1. e4 e5
2. Bc4 {Bishop's Opening.} Nf6 {Berlin Defense.}
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Center Game
(a) Paulsen Attack
1. e4 e5
2. d4 exd4 {Center Game.}
3. Qxd4 Nc6
4. Qe3 {Paulsen Attack.}
(b) Danish Gambit
1. e4 e5
2. d4 exd4 {Center Game.}
3. c3 {Danish Gambit.} d5 {Soerensen Defense.}
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Damiano's Defense
1. e4 e5
2. Nf3 f6?! {Damiano's Defense.}
3. Nxe5 fxe5??
4. Qh5+ {White wins.}
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Elephant Gambit
1. e4 e5
2. Nf3 d5 {Elephant Gambit.}
3. exd5 e4
4. Qe2 Nf6
5. Nc3 Be7
6. Nxe4
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Four Knights Game (1. e4 e5 2. Nf3 Nc6 3. Nc3 Nf6)
(Almost all reasonable continuations will suffice.)
Warning for Black: Don't steal White's e4-pawn before you're castled. (Re1 can pin your queen or knight against your king and win that piece.)
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King's Gambit (1. e4 e5 2. f4)
(Too heavily dependent on specific moves to summarize easily.)
Warning for White: Don't play fxe5. (That will allow Black to win with an attack starting with ...Qh4+.)
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Latvian Counter-gambit
1. e4 e5
2. Nf3 f5 {Latvian Counter-gambit.}
3. Nxe5 Qf6
4. d4 d6
5. Nc4
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Patzer Opening
1. e4 e5
2. Qh5 {Patzer Opening.} Nc6
3. Bc4 g6
4. Qf3 Nf6
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Petrov's Defense
(a) Main Line
1. e4 e5
2. Nf3 Nf6 {Petrov's Defense.}
3. Nxe5 d6
4. Nf3 {Main Line.} Nxe4
(b) Modern Attack
1. e4 e5
2. Nf3 Nf6 {Petrov's Defense.}
3. d4 {Modern Attack.} Nxe4 {Symmetrical Variation.}
(c) Petrov Three Knights Game
1. e4 e5
2. Nf3 Nf6 {Petrov's Defense.}
3. Nc3 {Petrov Three Knights Game.} Bb4
4. Nxe5 O-O
5. Be2 Re8
6. Nd3
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Philidor's Defense
(a) Exchange Variation
1. e4 e5
2. Nf3 d6 {Philidor's Defense.}
3. d4 exd4 {Exchange Variation.}
(b) Hanham Variation
1. e4 e5
2. Nf3 d6 {Philidor's Defense.}
3. d4 Nd7 {Hanham Variation.}
4. Bc4
(c) 'Wilson Defense'
1. e4 e5
2. Nf3 d6 {Philidor's Defense.}
3. d4 Bg4?! {'Wilson Defense.'}
4. dxe5 Bxf3
5. Qxf3 dxe5
6. Bc4 Nf6
7. Qb3 {White wins.}
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Ponziani Opening (1. e4 e5 2. Nf3 Nc6 3. c3)
(Almost all reasonable continuations will suffice.)
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Ruy Lopez (1. e4 e5 2. Nf3 Nc6 3. Bb5)
(Too heavily dependent on specific moves to summarize easily.)
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Scotch Opening
1. e4 e5
2. Nf3 Nc6
3. d4 {Scotch Opening.} exd4
4. Nxd4
Warning for Black: Don't play ...Nxd4. (That will allow White to centralize his queen.)
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other King's Pawn Openings
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Alekhine's Defense (1. e4 Nf6)
(Almost all reasonable continuations will suffice.)
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Caro-Kann Defense
1. e4 c6 {Caro-Kann Defense.}
2. d4 d5
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Fred
1. e4 f5 {Fred.}
2. exf5 Nf6
3. d4 d5
4. Bd3 Nc6
5. c3
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French Defense
1. e4 e6 {French Defense.}
2. d4 d5
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Modern Defense
1. e4 g6 {Modern Defense.}
2. d4 Bg7
3. Nc3
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Pirc Defense (1. e4 d6 2. d4 Nf6 3. Nc3)
(Almost all reasonable continuations will suffice.)
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Scandinavian Defense
(a) 'Anderssen Defense'
1. e4 d5 {Scandinavian Defense.}
2. exd5 Qxd5 {'Anderssen Defense.'}
3. Nc3 Qa5 {"Prie Defense."}
4. d4 Bf5 {"Besnyo Defense."}
5. Nf3
(b) Marshall Gambit
1. e4 d5 {Scandinavian Defense.}
2. exd5 Nf6 {Marshall Gambit.}
3. d4 {Modern Variation.} Nxd5 {Marshall Variation.}
4. c4
(c) "Vuckovic Attack"
1. e4 d5 {Scandinavian Defense.}
2. exd5 Qxd5 {'Anderssen Defense.'}
3. Nf3 {"Vuckovic Attack."} Nf6
4. d4
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Sicilian Defense
(a) with ...d6
1. e4 c5 {Sicilian Defense.}
2. Nf3 d6
3. d4 cxd4
4. Nxd4 Nf6
5. Nc3
Tip for Black: Don't play ...e5 when your pawn is at d6 unless you know what you're doing. (...e5 makes the d6-pawn backward and therefore likely to be lost, and leaves a hole at d5.)
(b) with ...e6
1. e4 c5 {Sicilian Defense.}
2. Nf3 e6 {Cramling Defense.}
3. d4 cxd4
4. Nxd4
(c) with ...Nc6
1. e4 c5 {Sicilian Defense.}
2. Nf3 Nc6
3. d4 cxd4
4. Nxd4 Nf6
5. Nc3
Tip for White: Don't play Nxc6. (That gives Black a useful tempo upon recapture by a pawn.}
(d) Smith-Morra Gambit
1. e4 c5 {Sicilian Defense.}
2. d4 {Smith-Morra Gambit.} cxd4
3. c3 {"Zelic Attack."} dxc3
4. Nxc3 Nc6
Warning for Black: Don't play ...Nxd4. (That will allow White to centralize his queen.)
(e) Alapin's Variation
1. e4 c5 {Sicilian Defense.}
2. c3 {Alapin's Variation.} d5
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Three Knights Game (1. e4 e5 2. Nf3 Nc6 3. Nc3 non-...Nf6)
(Almost all reasonable continuations will suffice.)
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Vienna Game
1. e4 e5
2. Nc3 {Vienna Game.} Nf6 {Falkbeer Variation.}
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Queen's Pawn Opening (1. d4)
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Queen's Pawn Game (1. d4 d5)
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Blackmar Gambit
1. d4 d5
2. e4 {Blackmar Gambit.} dxe4
3. Nc3 Nf6
4. f3 exf3
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Chigorin Variation
1. d4 d5
2. Nc3 {Chigorin Variation.} Nf6
3. Bg5 {Richter-Veresov Attack.} Nbd7
4. Nf3 h6
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Colle System (1. d4 d5 2. Nf3 Nf6 3. e3 e6 4. Bd3 c5 5. c3)
(Almost all reasonable continuations will suffice.)
Tip for Black: Leave your c5-pawn there until White captures it; don't play ...cxd4 or ...c4. (Retains tension, avoids useless advance.)
Tip for Black: It doesn't matter whether your KB is placed at d6 or e7. (After dxc5 ...Bxc5 the bishop will end up at c5 anyway.)
Tip for both sides: Castle kingside. (This is standard and probably safest.)
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Mason Variation
1. d4 d5
2. Bf5 {Mason Variation.} Nf6
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Queen's Gambit Accepted
1. d4 d5
2. c4 {Queen's Gambit.} dxc4 {Queen's Gambit Accepted.}
3. Nf3
Warning for Black: Don't try to retain your extra pawn (at c4)! (You will destroy your position trying to do so.)
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Queen's Gambit Declined (1. d4 d5 2. c4 non-...dxc4)
(Almost all reasonable continuations will suffice.)
Tip for Black: Don't block your c-pawn with your QN. (That will render your QN largely useless.)
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Indian Game (1. d4 Nf6)
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Bogo-Indian Defense (1. d4 Nf6 2. c4 e6 3. Nf3 Bb4+)
(Almost all reasonable continuations will suffice.)
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Budapest Defense
1. d4 Nf6
2. c4 e5 {Budapest Defense Declined.}
3. dxe5 Ng4 {Budapest Defense.}
4. Nf3 {Adler Variation.} c5
5. e3 Nc6
6. Be2 Ngxe5
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King's Indian Defense
1. d4 Nf6
2. c4 g6 {King's Indian Defense.}
3. Nc3 Bg7
4. e4 d6
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Nimzo-Indian Defense (1. d4 Nf6 2. c4 e6 3. Nc3 Bb4)
(Almost all reasonable continuations will suffice.)
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Queen's Indian Defense (1. d4 Nf6 2. c4 e6 3. Nf3 b6)
(Almost all reasonable continuations will suffice.)
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other Queen's Pawn Openings
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Dutch Defense (1. d4 f5)
(Almost all reasonable continuations will suffice.)
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other openings
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Bird's Opening (1. f4)
(Almost all reasonable continuations will suffice.)
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Dunst Opening
1. Nc3 {Dunst Opening.} d5
2. e4 d4
3. Nce2 e5
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English Opening (1. c4)
(Almost all reasonable continuations will suffice.)
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Grob's Attack
1. g4 {Grob's Attack.} d5
2. Bg2 c6
3. h3 e5
Tip for Black: Play ...Ne7, not ...Nf6. (White will chase off the knight via g5 if the knight develops to f6.}
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Polish Opening (1. b4)
(Almost all reasonable continuations will suffice.)
Tip for Black: Play ...Nbd7, not ...Nc6. (White will chase off the knight via b5 if the knight develops to c6.}
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Zukertort Opening (1. Nf3)
(Almost all reasonable continuations will suffice.)
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You're welcome. By the way I just made a few minor clarifications and repasted it. It's nice to see somebody give a thank-you these days.