  Let $F_2$ denote the free group on two generators $a,b$. For any
measure-preserving system $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ on a
finite measure space $X = (X,{\mathcal X},\mu)$, any $f \in L^1(X)$, and any $n
\geq 1$, define the averaging operators $${\mathcal A}_n f(x) := \frac{1}{4
\times 3^{n-1}} \sum_{g \in F_2: |g| = n} f( T_g^{-1} x ),$$ 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.
