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).

  Expected halting time for "The 2^n 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).