I learned about the "secretary problem" when I was about 10 years old from one of Martin Gardner's books. Though I thought is was cool and amazing, I don't think it gave me much insight into $e$.
Here's a way to introduce $e$ with only addition and multiplication, in the form of a game.
Tell him he's got a "budget" of say, 100 to work with, and his goal is to pick a bunch of (positive) numbers, not necessarily whole numbers, with that add up to 100, where he tries to make the product being as large as possible.
In his mind, he might first think to break his 100 as 50$\times$50, then realize that 25$\times$25$\times$25$\times$25 is even better, then 10$\times$10$\times$10$\times$10$\times$10$\times$10$\times$10$\times$10$\times$10$\times$10 is even better, and so on. The more numbers you split it into, the better!