  Two $n \times n$ Latin squares $L_1, L_2$ are said to be orthogonal if, for
every ordered pair $(x,y)$ of symbols, there are coordinates $(i,j)$ such that
$L_1(i,j) = x$ and $L_2(i,j) = y$. A $k$-MOLS is a sequence of $k$
pairwise-orthogonal Latin squares, and the existence and enumeration of these
objects has attracted a great deal of attention.
  Recent work of Keevash and Luria provides, for all fixed $k$,
log-asymptotically tight bounds on the number of $k$-MOLS. To study the
situation when $k$ grows with $n$, we bound the number of ways a $k$-MOLS can
be extended to a $(k+1)$-MOLS. These bounds are again tight for constant $k$,
and allow us to deduce upper bounds on the total number of $k$-MOLS for all
$k$. These bounds are close to tight even for $k$ linear in $n$, and readily
generalize to the broader class of gerechte designs, which include Sudoku
squares.
