It stands for entertaiment, apparently, in this quite fantastic training video for being a walkabout cartoon character.
See web-only content:
http://www.theatlantic.com/daily-dish/archive/2007/09/the-e-in-chuck-e-cheese/224978/
This article available online at:
http://www.theatlantic.com/daily-dish/archive/2007/09/the-e-in-chuck-e-cheese/224978/
