user16321: Let's say it's the last day of a tournament and white needs to win a game in order to advance to the next stage or something like that. Unfortunately, he reaches a position where there is no way to progress without making things worse somehow, and so he begins repeating the same position.
However, he doesn't want to claim a draw since it would ruin the tournament for him. Black realizes what's going on and also refuses to claim a draw, hoping that eventually white will give in and make an inferior move in an attempt to win.
Would black's behavior be considered unsporting in such a situation? Has something like this happened in a tournament before? Could this theoretically go on forever in a game with time increment?

user12629: I don't think this is unsporting, unless it lasts so long that it delays the tournament. White has the option of claiming a draw, and he can do so at any time. White also has the option of playing an inferior move and trying to win at all costs.  Repeating the position multiple times does nothing but delay this choice.

Has something like this happened in a tournament before?

Multiple repetition without a draw happened in one of the first strong tournaments! I don't know which round this game was played in, but as you can see, both players repeat the position many times, at multiple points in the game - and then one of them eventually plays another move. Look at what happens around moves 17, 33, and 86.
[Event "Baden-Baden"]
[Date "1870.??.??"]
[White "Neumann, G"]
[Black "Steinitz, W"]
[Result "0-1"]
[FEN ""]

1.d4 f5 2.e4 2...fxe4 3.Nc3 Nf6 4.Bg5 c6 5.Bxf6 exf6 6.Nxe4 d5 7.Ng3 Qb6 8.Qe2+ Kf7 9.O-O-O Na6 10.Qf3 g6 11.Bd3 Nb4 12.Kb1 h5 13.h3 h4 14.N3e2 Nxd3 15.Rxd3 Bf5 16.Rb3 Qc7 17.Nf4 Be4 18.Qg4 Bf5 19.Qf3 Be4 20.Qg4 Bf5 21.Qf3 Be4 22.Qg4 Bf5 23.Qf3 Be4 24.Qg4 Bf5 25.Qf3 Be4 26.Qg4 Bf5 27.Qf3 Be4 28.Qg4 Bf5 29.Qf3 Bd6 30.Nge2 a5 31.Rg1 a4 32.Re3 Rae8 33.g3 Be4 34.Qg4 Bf5 35.Qf3 Be4 36.Qg4 Bf5 37.Qf3 Be4 38.Qg4 Bf5 39.Qf3 Be4 40.Qg4 Bf5 41.Rxe8 Rxe8 42.Qf3 Qa5 43.Kc1 Qa6 44.Nc3 b5 45.g4 Bc8 46.Nd1 b4 47.Ne3 b3 48.a3 bxc2 49.g5 f5 50.Nxc2 Qc4 51.Ng2 Re2 52.Ne1 f4 53.Kd1 Re4 54.Ng2 Qb3 55.Qc3 Qxc3 56.bxc3 Bxh3 57.f3 Re8 58.Nxh4 Rh8 59.Ng2 Rh5 60.Nge1 Bf5 61.Nb4 Bd7 62.Rg2 Ke6 63.Ned3 Kf5 64.Nb2 Rh1+ 65.Kc2 Ra1 66.Nxa4 Rxa3 67.Nc5 Bxc5 68.dxc5 d4 69.cxd4 Rxf3 70.d5 Rg3 71.Rf2 f3 72.dxc6 Be6 73.Kd2 Ke4 74.Rf1 Rg2+ 75.Kc3 Re2 76.c7 Re3+ 77.Kd2 Re2+ 78.Kd1 Bb3+ 79.Kc1 Be6 80.Nc2 Bc8 81.Kd1 Bg4 82.c6 Kf4 83.Nd4 Re8 84.Kd2 Rc8 85.Kd3 Kg3 86.Rg1+ Kf4 87.Rf1 Kg3 88.Rg1+ Kf4 89.Rf1 Kg3 90.Rg1+ Kf4 91.Rf1 Kg3 92.Rg1+ Kf4 93.Rf1 Kg3 94.Rg1+ Kf4 95.Rf1 Kg3 96.Rg1+ Kf4 97.Rf1 Kg3 98.Rg1+ Kf4 99.Rf1 Kg3 100.Rg1+ Kf4 101.Rf1 Re8 102.Kc4 Kg3 103.Nxf3 Bxf3 104.Rxf3+ Kxf3 105.Kd5 Ra8 106.Ke5 Ke3 107.Kf6 Rc8 108.Kxg6 Kf4 109.Kf6 Rxc7 110.g6 Rxc6+ 111.Kf7 Kf5 112.g7 Rc7+ 113.Kf8 Kf6 114.g8=N+ Ke6 115.Nh6 Rh7 116.Ng4 Rh4 117.Ne3 Re4 118.Nd1 Rf4+ 119.Kg7 Rf3 120.Kg6 Ke5 121.Kg5 Kd4 122.Kg4 Rf1 123.Nb2 Rb1 124.Na4 Rb4 0-1


user4418: 
Could this theoretically go on forever in a game with time increment?

No, because the arbiter would step in and declare the game drawn after the fifth repetition of the position.
According to the FIDE Laws of Chess -

9.6 If one or both of the following occur(s) then the game is drawn:
  9.6.1 the same position has appeared, as in 9.2.2 at least five times.

and -

9.2.1 The game is drawn, upon a correct claim by a player having the move, when the same position for at least the third time (not
  necessarily by a repetition of moves):
  9.2.1.1 is about to appear, if he first writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his
  intention to make this move, or
  9.2.1.2 has just appeared, and the player claiming the draw has the move.


16321: <p>Let's say it's the last day of a tournament and white needs to win a game in order to advance to the next stage or something like that. Unfortunately, he reaches a position where there is no way to progress without making things worse somehow, and so he begins repeating the same position.</p>

<p>However, he doesn't want to claim a draw since it would ruin the tournament for him. Black realizes what's going on and also refuses to claim a draw, hoping that eventually white will give in and make an inferior move in an attempt to win.</p>

<p>Would black's behavior be considered unsporting in such a situation? Has something like this happened in a tournament before? Could this theoretically go on forever in a game with time increment?</p>
 12629: <p>I don't think this is unsporting, unless it lasts so long that it delays the tournament. White has the option of claiming a draw, and he can do so at any time. White also has the option of playing an inferior move and trying to win at all costs.  Repeating the position multiple times does nothing but delay this choice.</p>

<blockquote>
  <p>Has something like this happened in a tournament before?</p>
</blockquote>

<p>Multiple repetition without a draw happened in one of the first strong tournaments! I don't know which round this game was played in, but as you can see, both players repeat the position many times, at multiple points in the game - and then one of them eventually plays another move. Look at what happens around moves 17, 33, and 86.</p>

<pre><code>[Event "Baden-Baden"]
[Date "1870.??.??"]
[White "Neumann, G"]
[Black "Steinitz, W"]
[Result "0-1"]
[FEN ""]

1.d4 f5 2.e4 2...fxe4 3.Nc3 Nf6 4.Bg5 c6 5.Bxf6 exf6 6.Nxe4 d5 7.Ng3 Qb6 8.Qe2+ Kf7 9.O-O-O Na6 10.Qf3 g6 11.Bd3 Nb4 12.Kb1 h5 13.h3 h4 14.N3e2 Nxd3 15.Rxd3 Bf5 16.Rb3 Qc7 17.Nf4 Be4 18.Qg4 Bf5 19.Qf3 Be4 20.Qg4 Bf5 21.Qf3 Be4 22.Qg4 Bf5 23.Qf3 Be4 24.Qg4 Bf5 25.Qf3 Be4 26.Qg4 Bf5 27.Qf3 Be4 28.Qg4 Bf5 29.Qf3 Bd6 30.Nge2 a5 31.Rg1 a4 32.Re3 Rae8 33.g3 Be4 34.Qg4 Bf5 35.Qf3 Be4 36.Qg4 Bf5 37.Qf3 Be4 38.Qg4 Bf5 39.Qf3 Be4 40.Qg4 Bf5 41.Rxe8 Rxe8 42.Qf3 Qa5 43.Kc1 Qa6 44.Nc3 b5 45.g4 Bc8 46.Nd1 b4 47.Ne3 b3 48.a3 bxc2 49.g5 f5 50.Nxc2 Qc4 51.Ng2 Re2 52.Ne1 f4 53.Kd1 Re4 54.Ng2 Qb3 55.Qc3 Qxc3 56.bxc3 Bxh3 57.f3 Re8 58.Nxh4 Rh8 59.Ng2 Rh5 60.Nge1 Bf5 61.Nb4 Bd7 62.Rg2 Ke6 63.Ned3 Kf5 64.Nb2 Rh1+ 65.Kc2 Ra1 66.Nxa4 Rxa3 67.Nc5 Bxc5 68.dxc5 d4 69.cxd4 Rxf3 70.d5 Rg3 71.Rf2 f3 72.dxc6 Be6 73.Kd2 Ke4 74.Rf1 Rg2+ 75.Kc3 Re2 76.c7 Re3+ 77.Kd2 Re2+ 78.Kd1 Bb3+ 79.Kc1 Be6 80.Nc2 Bc8 81.Kd1 Bg4 82.c6 Kf4 83.Nd4 Re8 84.Kd2 Rc8 85.Kd3 Kg3 86.Rg1+ Kf4 87.Rf1 Kg3 88.Rg1+ Kf4 89.Rf1 Kg3 90.Rg1+ Kf4 91.Rf1 Kg3 92.Rg1+ Kf4 93.Rf1 Kg3 94.Rg1+ Kf4 95.Rf1 Kg3 96.Rg1+ Kf4 97.Rf1 Kg3 98.Rg1+ Kf4 99.Rf1 Kg3 100.Rg1+ Kf4 101.Rf1 Re8 102.Kc4 Kg3 103.Nxf3 Bxf3 104.Rxf3+ Kxf3 105.Kd5 Ra8 106.Ke5 Ke3 107.Kf6 Rc8 108.Kxg6 Kf4 109.Kf6 Rxc7 110.g6 Rxc6+ 111.Kf7 Kf5 112.g7 Rc7+ 113.Kf8 Kf6 114.g8=N+ Ke6 115.Nh6 Rh7 116.Ng4 Rh4 117.Ne3 Re4 118.Nd1 Rf4+ 119.Kg7 Rf3 120.Kg6 Ke5 121.Kg5 Kd4 122.Kg4 Rf1 123.Nb2 Rb1 124.Na4 Rb4 0-1
</code></pre>
 4418: <blockquote>
  <p>Could this theoretically go on forever in a game with time increment?</p>
</blockquote>

<p>No, because the arbiter would step in and declare the game drawn after the fifth repetition of the position.</p>

<p>According to the <a href="https://www.fide.com/fide/handbook.html?id=208&amp;view=article" rel="noreferrer">FIDE Laws of Chess</a> -</p>

<blockquote>
  <p>9.6 If one or both of the following occur(s) then the game is drawn:<br>
  9.6.1 the same position has appeared, as in 9.2.2 at least five times.</p>
</blockquote>

<p>and -</p>

<blockquote>
  <p>9.2.1 The game is drawn, upon a correct claim by a player having the move, when the same position for at least the third time (not
  necessarily by a repetition of moves):<br>
  9.2.1.1 is about to appear, if he first writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his
  intention to make this move, or<br>
  9.2.1.2 has just appeared, and the player claiming the draw has the move.</p>
</blockquote>
