It's been weeks (months?) since the 2048 game--by Gabriele Cirulli--took Internet by storm. I have an explicit integer $X$ which is greater or equal than any score of this game. Possibly my $X$ is the actual maximal score, I don't know, it may go either way. Thus my question: what is the maximal possible score for game 2048? (I'll provide $X$ in an answer below, as a SPOILER; it was quite straightforward to get $X$, and similar upper bounds for a way larger family of similar games). If $X$ is not sharp than sharper bounds are welcome too.

BTW, have anybody published an upper bound for 2048 on Internet or elsewere? (If there is a respective link then I'd like to see the article).

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**REMARK:** *While the earlier nice post:*

&nbsp; http://mathoverflow.net/questions/160703/expected-halting-time-for-the-2n-game-aka-2048-with-random-moves

*is related to the game 2048, this post and my question and my answer below are otherwise **not** related. It's nice to know about both posts, but that's all.*

*The other two posts mentioned provided a construction but not a **proof** that it provides maximum, and there is no other **proof** of a specific lower bound there. Their style seems to be of the type: I cannot do it better (without even attempting a mathematical proof of a bound). Please, show me in those posts that it's otherwise (a feeling that a construction is the best is not a proof).*