The number we are considering is as follows:
$0.a_1 a_2 a_3 \cdots $, where $a_{2n-1}=(n)_{(2)}, a_{2n}=(n)_{(3)}.$
So, the number is $$0.(1)(1)(10)(2)(11)(10)(100)(11)(101)(12)\cdots.$$ Is the number irrational? Is the number normal? Is the number transcendental?
It's not rational because the decimals don't repeat -- far enough out there are arbitrarily long runs of decimals without any 2's, yet there are still infinitely many 2's.
It cannot be normal in base 10 either, because the limiting frequency if 7's in the decimal expansion is 0 where it should be 1/10 for a normal number. It might be normal in other bases.
Transcendental? Most likely, though I can't construct an argument for it right off the cuff.