This looks equivalent to base 26 number system with the digits a = 0, b = 1, c = 2, d = 3,.............., y = 24, z = 25.
So, I think you have made a slight mistake because the "number" next to zzzz should be baaaa, not aaaaa (as in decimal number system the number next to 9999 is 10000, not 00000).
[ In fact you also have written aaba next to aaaz. So, clearly the number, which comes just after a number ending with 'z', is a number ending with 'a' & the second last digit INCREASING BY 1 (i.e. b goes to c, p goes to q, z goes to a etc.). For example, the number next to ****pz is ****qa.
This also implies that it's baaaa which comes just after zzzz. ]
Now, decimal equivalent of aaaa is a×26³ + a×26² + a×26¹ + a×26⁰ = 0 [as a = 0]
[Obviously! 0000 must be 0.]
& decimal equivalent of zzzzzz is
z×26⁵ + z×26⁴ + z×26³ + z×26² + z×26¹ + z×26⁰
= 25×26⁵ + 25×26⁴ + 25×26³ + 25×26² + 25×26¹ + 25×26⁰ [as z = 25]
So, you just need to find how many numbers are there from 0 to 308,915,775.
Now, that's really a piece of a cuppycake. You got the answer. Yes, it's 308,915,776.