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Find an irrational number that does not contain strings of $0$s of arbitrary length in its decimal representation.

I was wondering how to find such a number. Since it is irrational, the decimal cannot repeat. How do we make sure it does not contain strings of $0$s of arbitrary length in decimal notation?

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    Hint: you can build an irrational number using just digits $1$ and $2$ as long as you make sure that the decimal expansion is not periodic. – 2017-01-03
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    Don't use any zero digits! – 2017-01-03

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How about $$0.121122111222111122221111122222\dots$$ where the digits come in alternating blocks of $1$s and $2$s, with the lengths of the blocks increasing. It never repeats, and contains nothing but $1$s and $2$s.

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    And we can even avoid arbitrarily long blocks of *any* digit: e.g. $$0.122121121212212121211212121212...$$ (that is, $12$, $2121$, $121212$, $21212121$, . . . ). – 2017-01-03