I'll speak about strange phenomena that appear when we try to
multiply matrices. Setting: given a finite set $A$ of matrices, consider all
the possible products $A_1\cdot\ldots\cdot A_n$
with $A_i\in A$. How fast could
the norm of this product increase with $n$? How slow could it increase?
And how many infinite sequences of matrices from $A$ achieve this maximal
or minimal growth (in the sense of entropy for the full shift on $A^\infty$)?
The answers (those that are known) are highly nontrivial even in $SL(2,R)$.