I don't necessarily believe this, but I'll play along.

To make it appear natural so we don't notice, death is the first thing that comes to mind. So pandemics, disasters and wars that kill off beings on a large scale to free up memory. A globe with limited surface area seems ideal to stick us on to begin with, with anything outside of that sphere virtually impossible to access. The size of Earth could have been chosen because it fits comfortably within the RAM limits. If Earth is pushing the RAM limits, each planet could be hosted on its own server. So if we someday colonized Mars or the moon, the trip between would be like a server transfer making the RAM issues for interplanetary colonization inconsequential.

If you want to really explore the fringes of this concept, maybe those in the simulation would see glitches that shouldn't happen if it starts running out of RAM. UFOs, shadows, or synchronicities could become commonplace. People could randomly go catatonic or experience amnesia if they're personally impacted. If it got out of control across the entire simulation, perhaps a hard reset would become necessary. It may even be a planned cycle of hard resets based on the anticipated maximum lifespan of the simulation before things start to get fucky due to memory errors. So power on = big bang, and hard reset something like big crunch or heat death of the universe.