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).