What you have there is a trancedental number; a number for which there is no variable polynomial equation with rational coefficients that has this number as a root. Trancedental numbers are always irrational (but not all irrational numbers are trancedental). Therefore, yes, your number is irrational.
The proof is in the construction; like the well-known trancedental numbers $\pi$ and $e$, your number is the asymptotic limit of the sum of an infinite series; in this case, the sum of a reducing fraction:
$\sum_{n=1}^{\infty} \dfrac{1}{10^{\dfrac{n(n+1)}{2}}}$
This is similar to the construction of the Champernowne constant which is proven transcedental. The sum is constructed such that no 2 terms ever modify the value of a decimal place of the same order of magnitude, very much like $C_{10}$, and so the number constantly increases but can never reach a rational sum, unlike the infinite sum of $\frac{1}{2^n}$.