My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer.

After tiring of memorizing $\pi$, he resolved to discover a new irrational whose expansion is easier to memorize. He invented (probably re-invented) the number $J$:

$$J = 6.12345678910111213141516171819202122\ldots$$

which clearly lets you name as many digits as you like pretty easily. He asked me if $J$ is transcendental just like $\pi$, and I said it must be but I didn't know for sure. Is there an easy way to determine this?

I can show that $\pi$ is transcendental (using Lindemann-Weierstrass) but it doesn't work for arbitrary numbers like $J$, I don't think.