[albertzeyer]

Yes, but its finite storage. (You could argue that with infinite float precision, you would could get infinite storage. Some old proofs that RNNs are as powerful as a Turing machine work that way. Of course, in practice, it's very much limited.)

Also, it's somewhat hard to train the network that it can remember much information. (Not sure if much was done to measure this, but from my gut feeling, I would say, a RNN layer with about 500 neurons can be trained with the standard methods to maybe store 10 bits of information effectively.) The problem is that it can become easily very unstable.

The LSTM cell is already somewhat better in this regard. But of course, this is still finite memory then, and you cannot have much more than 500-1000 LSTM cells in one layer, because training becomes too computational expensive then. (You could introduce some bottleneck projection layer as Google did recently, and then get it to maybe 2000 LSTM cells.) Maybe count one LSTM cell as one bit of information (again, this is very much taken out of air).

This is much less memory than what your PC has (which of course also has finite memory, but it's so huge that you can count it as infinite and as powerful as a Turing machine).