Is it possible to prove that a game(ex. chess or othello...) will eventually halt at some point.
You can make games that will halt and those that won't.  Othello will always halt because it cannot continue for more than 60 moves based on the rules.  Chess ends only because there are rules to prevent infinite-move games (repetition rule, 50-move rule, etc).  You can certainly construct games that never end... a silly example is "whoever names the highest integer wins".

Then there are games that lead to a paradox when you try and ask if they will end or not.  A famous example is the game called CHOICE.  The first move of CHOICE is to choose a game that does eventually end.  The 2nd move of CHOICE is to play the first move of the chosen game, etc.

Now we ask, "does CHOICE end or not?"  If you say it does, then the first move of CHOICE could be to choose CHOICE, and continuing this way leads to a never-ending game.  So suppose we say CHOICE does not end, thereby disqualifying it as a possibility for the first move.  Then clearly it DOES end.