There is and there isn’t. The case for there is would go like this: let’s say, our brain can store 1000 doodles in each cell and a further 5000 doodles in each distinct connection (that is, the wiring itself as information). If we consider all the permutations and combination of cells and wiring, we get 10 raised to the power of some-doodillion. So there, that is the limit. Great. However, I think arriving at a number like this is questionable. It assumes that we have a valid definition of what memory is, which may not be true. (There are valid and verifiable definitions for computer memory but we are talking of human brain here).
To the next case: there is no limit to how much a human brain can store (and recall, because otherwise what’s the point). The case for this may be stated like this: What do we conceive of as unit memory? Let’s suppose that I don’t remember who I saw last month, but remember I did see someone whom I know. In this case how do I quantify this particular memory? Memory could be of real things and could be of other memories, it could be of memories of memories, and so on and so on – an infinite recursion like reflections between two mirrors. In this sense, there is no limit to our memory. I am, of course, being intentionally vague here. That’s because I am setting things up for the below.
There is a third case: the impossibility of quantifying memory. This can be stated in a more substantial and non-trivial way (and I will not be the one to do it, you give a try). The computational approach to memory – calculating storage space for unit memory, in my opinion, is misleading. Memory, it seems, needs characterization not quantification. But, what other approach is there? I don’t know. I am dissatisfied with all the ‘answers’ I’ve read on this. I was thinking about this yesterday night while going to sleep and confused myself terribly, once again.
Listen to Seth Lloyd for an overview of how one can frame a related question – that of processing limit.