  We analyse large deviations of time-averaged quantities in stochastic
processes with long-range memory, where the dynamics at time t depends itself
on the value q_t of the time-averaged quantity. First we consider the elephant
random walk and a Gaussian variant of this model, identifying two mechanisms
for unusual fluctuation behaviour, which differ from the Markovian case. In
particular, the memory can lead to large deviation principles with reduced
speeds, and to non-analytic rate functions. We then explain how the mechanisms
operating in these two models are generic for memory-dependent dynamics and
show other examples including a non-Markovian symmetric exclusion process.
