Successor Function in high numbers

Question

Recently, I tried creating a Peano-based googology model. I started with 10^3003, which I understand is named a millillion. I then defined a millillillion - or, for short, a thousand 2ill (there are 2 ills, because mille means thousand) - as 10^millillion. I then defined a thousand 3ill, then so on for a thousand nill. When you reach a thousand millillionill, 10^thousand millillionill is a first-order 1. The first-order successor function is 10^x. So first-order 2 is 10^first-order 1, etc. We'll use 1o for first-order to be short. 1o addition is defined as '1ox + 1oy = 10^10...^10^1oy, where there are x 10s. 1o multiplication is defined to the same way as 1o addition as normal multiplication is to normal addition. In the same way, we define 1o exponentation, etc. So we then reach, using the same method as above, 1o thousand millillionill, then 10^1o thousand millillionill = 2o1. Generally, the xo successor function is (x-1)o 10^xoy. So to get to, say, 100th-order 1 to 100th-order 2, you do 99o10^100o1. We can then reach a millillionth-order 1, then a thousand millillionillth-order 1. We can then reach second-period first-order 1, which is the number after a thousand millillionillth-order 1. My problem is that, since xo addition changes, I can't define how to get from xp1 to (x+1)p1. Please comment if you don't understand. Can somebody figure out the generalised formula for getting from xp1 to (x+1)p1?