One of my students asked me this, and it occurred to me that I had never really questioned it.
Apparently, it is only conjectured but widely believed that the decimal expansion in base $10$ of $\pi$ contains all finite strings of the numerals $0$ through $9$.
Am I even accurate that the conjecture is widely accepted? Regardless, what is the rationale for this belief? Do any good heuristics exist? It seems perfectly logical (dare I say likely) that, just maybe, the string 2347529384759748975847523462346435664900060906, for example, never occurs. The conjecture seems absurdly strong, to me.
And just because a separate question would be ridiculous: does this conjecture extend to other famous transcendental numbers? Is it indeed conjectured that this is a property of transcendental numbers in general?