Ok so my goal with this mechanic is to have any dice roll have any number between zero and infinity be roll able, and the average be controlled by the players score (stat + skill).

You have a score (Stat + skill), you pick up that many dice of any size or combination of sizes from D4, D6, D8, D10 and D12.  roll them.
All 1s you rolled on the dice add 0 to your result, all dice that rolled between 2 and the dices maximum number add 1 to your result, and all dice that rolled the highest number that dice can roll adds 1 to your result and allows you to roll an additional dice of the same size and add it to your dice pool.

my theory is that regardless of the dice size the average will will always be 1 per dice.  the size will just change the pitch of the bell curve.  I do not have the math skills to prove this, and my friends have found ways of showing its anything from 0.99999 to 1.8.  so i turn to the power of the internets.

if my discription confused you here is a cheet sheet
Dice/add 0/add 1/add1 and another dice
D4/1/2-3/4
D6/1/2-5/6
D8/1/2-7/8
D10/1/2-9/0 or 10
D12/1/2-11/12

Yes, the expected value for this mechanic is in fact 1 regardless of the used die. To prove that, you can just substitute the additional die with a flat one and calculate the expeced value that (one). However, since the operator for the expected value is linear, you can exchance any constant expression with another roll that has the same expected value. So if you define a roll D4* = {0, 1, 1, 2} and know the expected value of that is one, you can be sure that D4 = {0, 1, 1, 1+D4*} also has an expected value of one. This way, you can define a series with a finite number of D4-rolls before you break with a constant number of two instead of 1+D4, and the expected value will be one for all of them. Now, you can move that Limit out to infinity. A very similar proof can be formulated for all other dice.

Now, something about the implications of this mechanic:
I once had the same idea... Every value from zero to plus infinity should be rollable. However, that changed after a while. Specifically, after I had formulated some of the details of my system, and then realized that it would break down if people started rolling anything significantly bigger than 100, in a system where 20 would be an average roll and 40 would be a very good roll. So I changed it to have a fixed "maximum roll" for any given stat value.