Do memories have mass?
You would need to live a long, long, long, long time for this to become remotely problematic.

Your question seems to suppose that a memory is "stored" by a neuron, and since neurons have mass, then the more memories we have the more our brains will weigh. Actually, neurogenesis is pretty rare in the adult brain--most of the cortex is fixed, and new neurons do not grow. (The hippocampus is the best known counterexample, but it's important to note that contrary to lore, the hippocampus doesn't store memories per sé, though it is involved in the formation and retrieval of memories).

Memories are represented in the brain as patterns of firing neurons. Let's say, for simplicity, that each neuron can either be on or off. Since there are 86 billion neurons in the brain, we can experience 2^86bn possible brain states. That's a lot.

Of course there are a lot of simplifications here--most notably, there is a lot of structure to the brain, and so i'm not suggesting you can store 2^86bn memories--but the logic is the same. Even with only 1,000 neurons, we can store a ridiculous amount of information (2^1000).

I would like to comment on MCM's answer as well, because I don't think it's true that we "remove" old memories. Or rather, the topic is still being debated in cognitive science today. For an old take on it, see renowned memory researcher Endel Tulving's (1974) article, in which he says:


  When we forget something we once knew, it does not necessarily mean that the memory trace has been lost; it may only be inaccessible


The idea is that some memories are suppressed (no, not repressed), being rendered inaccessible. Robert Bjork and John Anderson have (separately) done some nice work in this area.

So before you worry about this becoming a problem, I suspect you need to work out the secret of immortality first.


  Tulving, E. (1974). Cue-Dependent Forgetting: When we forget something
  we once knew, it does not necessarily mean that the memory trace has
  been lost; it may only be inaccessible. American Scientist, 74-82.