In game theory, such sequences are needed all the time, and the notation $x_{-i}$ has become so common that it is often not even defined in papers.

The reason is that much of game theory is concerned with situations where each player $j$ has a presupposed strategy $x_j$ and we think of one player $i$ deviating from his given strategy to some other strategy $y_i$, while the other players do not deviate. This new outcome is often denoted by $(y_i,x_{-i})$ or $(y_i; x_{-i})$ or some such abuse of notation, instead of the cumbersome $(x_1,\ldots, x_{i-1},y_i,x_{i+1},\ldots,x_n)$. Despite the fact that "the indices are out of order," it is very convenient notation for game theory once you get used to it.

In particular it allows one to write conditions like $u_i(x)\geq u_i(y_i,x_{-i})$ for all players $i$ and all $y_i$ to define what it means for $x$ to be a Nash equilibrium. Other solution concepts can also be defined compactly with this notation.