Given $n$. Two players in turn write different real numbers $x_1,x_2,x_3,\dots$ The player after whose turn there is a monotone subsequence of length $n$ loses. I guess that the question 'who wins' may be hopeless (nice if not so), but possibly there exists a better estimate of the number of turns than Erdős–Szekeres bound $(n-1)^2+1$ (which works not only for optimal, but for all strategies)?
As noted in the comments (but with not quite the right reference) the game is a first player win for $n \geq 4$. The question here is about the misere form, so this is a combination of Proposition 7, Theorem 10 and a bit of Proposition 9 in the paper Monotonic Sequence Games by Albert (who he?), Aldred, Atkinson, Handley, Holton, McCaughan and Sagan. This also appeared in Games of no chance 3. We did not address the question of the length of the game when one player is playing using a winning strategy, and the other to lose as slowly as possible (obviously, if the players cooperate to lengthen the game then it reaches the Erdos-Szekeres bound). The proof uses a "strategy stealing" argument, but one which is not entirely clear cut. In common with most such arguments it does not actually yield a strategy for the first player to win, only a proof that such a strategy exists.