We investigate the effective memory depth of RNN models by using them for $n$-gram language model (LM) smoothing. Experiments on a small corpus (UPenn Treebank, one million words of training data and 10k vocabulary) have found the LSTM cell with dropout to be the best model for encoding the $n$-gram state when compared with feed-forward and vanilla RNN models. When preserving the sentence independence assumption the LSTM $n$-gram matches the LSTM LM performance for $n=9$ and slightly outperforms it for $n=13$. When allowing dependencies across sentence boundaries, the LSTM $13$-gram almost matches the perplexity of the unlimited history LSTM LM. LSTM $n$-gram smoothing also has the desirable property of improving with increasing $n$-gram order, unlike the Katz or Kneser-Ney back-off estimators. Using multinomial distributions as targets in training instead of the usual one-hot target is only slightly beneficial for low $n$-gram orders. Experiments on the One Billion Words benchmark show that the results hold at larger scale: while LSTM smoothing for short $n$-gram contexts does not provide significant advantages over classic N-gram models, it becomes effective with long contexts ($n > 5$); depending on the task and amount of data it can match fully recurrent LSTM models at about $n=13$. This may have implications when modeling short-format text, e.g. voice search/query LMs. Building LSTM $n$-gram LMs may be appealing for some practical situations: the state in a $n$-gram LM can be succinctly represented with $(n-1)*4$ bytes storing the identity of the words in the context and batches of $n$-gram contexts can be processed in parallel. On the downside, the $n$-gram context encoding computed by the LSTM is discarded, making the model more expensive than a regular recurrent LSTM LM.