Let
$F_{2}$
denote the free group on two generators
$a$
and
$b$
. For any measure-preserving system
$(X,{\mathcal{X}},{\it\mu},(T_{g})_{g\in F_{2}})$
on a finite measure space
$X=(X,{\mathcal{X}},{\it\mu})$
, any
$f\in L^{1}(X)$
, and any
$n\geqslant 1$
, define the averaging operators
$$\begin{eqnarray}\displaystyle {\mathcal{A}}_{n}f(x):=\frac{1}{4\times 3^{n-1}}\mathop{\sum }_{g\in F_{2}:|g|=n}f(T_{g}^{-1}x), & & \displaystyle \nonumber\end{eqnarray}$$
where
$|g|$
denotes the word length of
$g$
. We give an example of a measure-preserving system
$X$
and an
$f\in L^{1}(X)$
such that the sequence
${\mathcal{A}}_{n}f(x)$
is unbounded in
$n$
for almost every
$x$
, thus showing that the pointwise and maximal ergodic theorems do not hold in
$L^{1}$
for actions of
$F_{2}$
. This is despite the results of Nevo–Stein and Bufetov, who establish pointwise and maximal ergodic theorems in
$L^{p}$
for
$p>1$
and for
$L\log L$
respectively, as well as an estimate of Naor and the author establishing a weak-type
$(1,1)$
maximal inequality for the action on
$\ell ^{1}(F_{2})$
. Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator.