  The celebrated Oseledets theorem \cite{O}, building over seminal works of
Furstenberg and Kesten on random products of matrices and random variables
taking values on non-compact semisimple Lie groups \cite{FK,Furstenberg},
ensures that the Lyapunov exponents of $\mathrm{SL}(d,\mathbb R)$-cocycles
$(d\geqslant 2)$ over the shift are well defined for all points in a total
probability set, ie, a full measure subset for all invariant probabilities.
Given a locally constant $\mathrm{SL}(d,\mathbb R)$-valued cocycle we are
interested both in the set of points on the shift space for which some Lyapunov
exponent is not well defined, and in the set of directions on the projective
space $\mathbf P \mathbb R^d$ along which there exists no well defined
exponential growth rate of vectors for a certain product of matrices. We prove
that if the semigroup generated by finitely many matrices in
$\mathrm{SL}(d,\mathbb R)$ is not compact and is strongly projectively
accessible then there exists a dense set of directions in $\mathbf P \mathbb
R^d$ along which the Lyapunov exponent of a typical product of such matrices is
not well defined. As a consequence, we deduce that the presence of Lyapunov
non-typical behavior is prevalent among $SL(3,\mathbb R)$-valued hyperbolic
cocycles. These results arise as a consequence of the more general description
of the set of non-typical points for Ces\`aro averages of continuous
observables and infinite paths in finitely generated semigroup actions by
homeomorphisms on a compact metric space $X$ under a mild hitting times
condition.
