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).

PS:   I'd like to see a quote of an essential moment of a rigorous proof of an upper bound for a maximal tile.

I'll clarify the simple issue of an upper bound for the total sum:

Since I have my bound for the maximal tile, let me follow it with an upper bound for the total sum. Actually, it is obtained as corollary to a maximal configuration (even if such a configuration can be fictitious--nevertheless, all tiles have values not exceeding the respective tiles of the fictitious configuration).

Once again, it will be easier to consider the general templates (see the answer below) and the logarithmic notation.

Consider $\ A=A_0=A_1=\ldots$ and $\ \forall_{n=0\ 1\ \ldots}\ |A_n|=b\ $ be finite (the playing board is finite so-to-speak). Consider an arbitrary non-negative integer $n$. For each $\ x\in A_n\ $ consider one of its oldest ancestors. By permuting sets $\ A_n\ $ we may assume that $\ x\ $ itself is always its ancestor whenever there is an ancestor for $\ x.\ $ Now let $\ \xi\ $ be the set of all $\ y\in A_n\,\ $ different from $\ x,\ $ for which there is an ancestor at least as old as the one for $\ x.\\\$. Then $\ x\ $ has all its ancestors in $\ A(x) := A\setminus\xi.\ $ Observe that functions (*moves*) preserve $\ A(x).\ $ hence we can consider the induced template for $\ A(x)\ $ and the respective bound:

$$h_n(x)\ =\ (b-|\xi|) + S - 1$$

That's all.

REMARK The permutation argument is obvious in the context of templates while it feels messy in the special case of the game 2048.