(By continued fraction sequence, I'm specifically talking about the one kind where the numerator of every fraction is 1.)
As a kid in middle school, I learned that all irrational numbers have non-repeating, non-terminating positional notation (o.k.a. "decimal") expansions.
However, as a kid in first-year university, I learned that some irrational algebraic numbers have repeating continued fraction expansions (and, of course, that all rationals have a finite continued fraction expansion). So my question now is, do all algebraic numbers have a repeating continued fraction expansion? Are there some transcendentals that have a repeating expansion, or algebraic numbers that have a non-repeating expansion?
No, continued faction expansions repeat only for the so-called "quadratic irrationals", numbers $x$ satisfying $ax^2 + bx + c = 0$ for some integers $a, b$ and $c$. In particular, the continued fraction expansion for the algebraic number $\sqrt[3]2$ does not repeat.
Continued Fractions periodicity and convolution on this site has a proof.