user-1: Are there any recorded examples of two chess games that were identical till the end? 
If the answer is no (which is most likely the case), what is the longest number of moves that have been identical in two independently played games on record? 

user10295: There are Several reasons for identical games:

The fact that we databases and computers allows us to find the best move in known positions. (if both players knows the theory in the same level - it might lead to a known draw).

Hence some players might follow a known game between two great players, which might result with identical games.

The fact that there are known traps, might lead to to the same trap (when one player knows the theory and the other fall in the trap) 

In this case, the game might end after the player fall in the trap.

In same cases, following a theory might lead to a lose as in the below story about Vishy Anand:

Vishy Anand loses in 6 moves after following a known game
[White "Zapata"]
[Black "Anand"]

[FEN ""]

1.e4 e5 2.Nf3 Nf6 3.Nxe5 d6 4.Nf3 Nxe4 5.Nc3 Bf5?? 6.Qe2

Anand resigns

The story goes: reaching the position after white's 5th move,Anand
  recalled the publication of the game Miles-Christiansen,in the
  Informator(a highly respectable chess journal),which went 6.Nxe4,Bxe4
  etc....Impressed with the ease with which black had drawn in that
  game,Vishy decided to try it himself.But,after 6.Qe2,black just loses
  a piece. 
Now,how could both Miles and Christiansen have missed such an
  obvious move???? The answer is........they didn't!!That game had been
  agreed drawn in advance.Something the Informator had not revealed.At
  the board,Miles did see that Qe2 was winning,but remained the
  gentleman,and avoided playing it.It is said,that he did spend some
  seconds polishing the e2 square with his finger,untill Christiansen's
  face had the appropriate shade of red,whereupon Miles took on e4.

Two Identical games
Here you can find

Two identical games for the first 28 moves

A Lopukhin vs Alexander Semeniuk (1975)
Yakov Estrin vs Hans Berliner (1965)

48 Moves - Both Haugen (2523)-Fernandez (2437), 2010, and Bross (2587)-Efremov (2574), 2011, reached the same position after 48...Qd7+:

This is my source for the below Answer, in this link you can find more identical games.
[Event "Uxbridge Int Alekhine"]
[Site "Uxbridge ENG"]
[Date "2010.02.17"]
[Round "8"]
[White "Rudd, J."]
[Black "Gormally, D."]
[Result "1/2-1/2"]
[ECO "B90"]
[WhiteElo "2286"]
[BlackElo "2479"]
[PlyCount "18"]
[EventDate "2010.02.13"]
[EventType "tourn"]
[EventRounds "9"]
[EventCountry "ENG"]
[EventCategory "2"]
[Source "Mark Crowther"]
[SourceDate "2010.03.01"]

[FEN ""] 

1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Be3 Ng4 7. Bc1 Nf6 8.
Be3 Ng4 9. Bc1 Nf6 1/2-1/2

Another Game:
[Event "ch-SUI"]
[Site "Lenzerheide SUI"]
[Date "2010.07.16"]
[Round "9"]
[White "Jenni, F."]
[Black "Kurmann, O."]
[Result "1/2-1/2"]
[ECO "B90"]
[WhiteElo "2542"]
[BlackElo "2449"]
[PlyCount "18"]
[EventDate "2010.07.08"]
[EventRounds "9"]
[EventCountry "SUI"]
[Source "Mark Crowther"]
[SourceDate "2010.07.19"]

[FEN ""] 

1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Be3 Ng4 7. Bc1 Nf6 8.
Be3 Ng4 9. Bc1 Nf6 1/2-1/2

Another Game:
[Event "Moscow Dvorkovich Memorial"]
[Site "Moscow"]
[Date "2009.09.11"]
[Round "4"]
[White "Gansvind, Valeriya I"]
[Black "Pisakov, Ilia"]
[Result "1/2-1/2"]
[ECO "B90"]
[WhiteElo "2226"]
[BlackElo "2331"]
[PlyCount "18"]
[EventDate "2009.09.08"]
[EventType "tourn"]
[EventRounds "13"]
[EventCountry "RUS"]
[EventCategory "4"]
[Source "ChessBase"]
[SourceDate "2009.11.03"]

[FEN ""] 

1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Be3 Ng4 7. Bc1 Nf6 8.
Be3 Ng4 9. Bc1 Nf6 1/2-1/2


-1: <p>Are there any recorded examples of two chess games that were identical till the end? </p>

<p>If the answer is no (which is most likely the case), what is the longest number of moves that have been identical in two independently played games on record? </p>
 10295: <p><strong>There are Several reasons for identical games:</strong></p>

<ol>
<li>The fact that we databases and computers allows us to find the <em>best</em> move in known positions. (if both players knows the <em>theory</em> in the same level - it might lead to a known draw).</li>
</ol>

<p>Hence some players might follow a known game between two great players, which might result with identical games.</p>

<ol start="2">
<li>The fact that there are known traps, might lead to to the same trap (when one player knows the <em>theory</em> and the other fall in the trap) </li>
</ol>

<p>In this case, the game might end after the player fall in the trap.</p>

<ol start="3">
<li>In same cases, following a <em>theory</em> might lead to a lose as in the below story about Vishy Anand:</li>
</ol>

<p><a href="https://www.redhotpawn.com/forum/only-chess/vishy-loses-in-6-movesthe-story.19148" rel="noreferrer"><strong>Vishy Anand loses in 6 moves after following a known game</strong></a></p>

<pre><code>[White "Zapata"]
[Black "Anand"]

[FEN ""]

1.e4 e5 2.Nf3 Nf6 3.Nxe5 d6 4.Nf3 Nxe4 5.Nc3 Bf5?? 6.Qe2
</code></pre>

<p><strong>Anand resigns</strong></p>

<blockquote>
  <p>The story goes: reaching the position after white's 5th move,Anand
  recalled the publication of the game Miles-Christiansen,in the
  Informator(a highly respectable chess journal),which went 6.Nxe4,Bxe4
  etc....Impressed with the ease with which black had drawn in that
  game,Vishy decided to try it himself.But,after 6.Qe2,black just loses
  a piece. </p>
  
  <p>Now,how could both Miles and Christiansen have missed such an
  obvious move???? The answer is........they didn't!!<strong>That game had been
  agreed drawn in advance</strong>.Something the Informator had not revealed.At
  the board,Miles did see that Qe2 was winning,but remained the
  gentleman,and avoided playing it.It is said,that he did spend some
  seconds polishing the e2 square with his finger,untill Christiansen's
  face had the appropriate shade of red,whereupon Miles took on e4.</p>
</blockquote>

<p><strong>Two Identical games</strong></p>

<p><a href="https://www.quora.com/What-is-the-maximum-number-of-identical-moves-in-two-different-official-chess-games-between-grandmasters#!n=12" rel="noreferrer">Here you can find</a></p>

<ol>
<li><p>Two identical games for the first 28 moves</p>

<ul>
<li><a href="http://www.chessgames.com/perl/chessgame?gid=1225038" rel="noreferrer">A Lopukhin vs Alexander Semeniuk (1975)</a></li>
<li><a href="http://www.chessgames.com/perl/chessgame?gid=1224863" rel="noreferrer">Yakov Estrin vs Hans Berliner (1965)</a></li>
</ul></li>
<li><p>48 Moves - Both Haugen (2523)-Fernandez (2437), 2010, and Bross (2587)-Efremov (2574), 2011, reached the same position after 48...Qd7+:</p></li>
</ol>

<p>This is <a href="https://www.chess.com/forum/view/general/identical-chess-games-at-master-level" rel="noreferrer">my source</a> for the below Answer, in <a href="https://www.chess.com/forum/view/general/identical-chess-games-at-master-level" rel="noreferrer">this link</a> you can find more identical games.</p>

<pre><code>[Event "Uxbridge Int Alekhine"]
[Site "Uxbridge ENG"]
[Date "2010.02.17"]
[Round "8"]
[White "Rudd, J."]
[Black "Gormally, D."]
[Result "1/2-1/2"]
[ECO "B90"]
[WhiteElo "2286"]
[BlackElo "2479"]
[PlyCount "18"]
[EventDate "2010.02.13"]
[EventType "tourn"]
[EventRounds "9"]
[EventCountry "ENG"]
[EventCategory "2"]
[Source "Mark Crowther"]
[SourceDate "2010.03.01"]

[FEN ""] 

1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Be3 Ng4 7. Bc1 Nf6 8.
Be3 Ng4 9. Bc1 Nf6 1/2-1/2
</code></pre>

<p>Another Game:</p>

<pre><code>[Event "ch-SUI"]
[Site "Lenzerheide SUI"]
[Date "2010.07.16"]
[Round "9"]
[White "Jenni, F."]
[Black "Kurmann, O."]
[Result "1/2-1/2"]
[ECO "B90"]
[WhiteElo "2542"]
[BlackElo "2449"]
[PlyCount "18"]
[EventDate "2010.07.08"]
[EventRounds "9"]
[EventCountry "SUI"]
[Source "Mark Crowther"]
[SourceDate "2010.07.19"]

[FEN ""] 

1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Be3 Ng4 7. Bc1 Nf6 8.
Be3 Ng4 9. Bc1 Nf6 1/2-1/2
</code></pre>

<p>Another Game:</p>

<pre><code>[Event "Moscow Dvorkovich Memorial"]
[Site "Moscow"]
[Date "2009.09.11"]
[Round "4"]
[White "Gansvind, Valeriya I"]
[Black "Pisakov, Ilia"]
[Result "1/2-1/2"]
[ECO "B90"]
[WhiteElo "2226"]
[BlackElo "2331"]
[PlyCount "18"]
[EventDate "2009.09.08"]
[EventType "tourn"]
[EventRounds "13"]
[EventCountry "RUS"]
[EventCategory "4"]
[Source "ChessBase"]
[SourceDate "2009.11.03"]

[FEN ""] 

1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Be3 Ng4 7. Bc1 Nf6 8.
Be3 Ng4 9. Bc1 Nf6 1/2-1/2
</code></pre>
