FWIW, all of the convergents from the continued fraction expansion of a number have "quality" greater than 2. Conversely, the Thue–Siegel–Roth theorem says that any algebraic number and any epsilon > 0, the algebraic number has only finitely many approximations with quality more than 2 + epsilon; so the "quality" values of successive best approximations are a series which converges to 2 from above.
This was how the first known transcendental numbers -- the Liouville numbers -- were constructed: They have approximations with unbounded quality, thus they cannot be algebraic. In practice, however, this isn't a very useful method: "Almost all" transcendental numbers also obey the Thue–Siegel–Roth theorem.
Is it also the case that all transcendental numbers have approximations of unbounded quality? Or only that Liouville numbers have approximations of unbounded quality, and therefore cannot be algebraic?
Liouville numbers have approximations of unbounded quality and thus cannot be algebraic. "Most" transcendental numbers only have finitely many approximations of quality higher than 2 + epsilon for any epsilon.
This was how the first known transcendental numbers -- the Liouville numbers -- were constructed: They have approximations with unbounded quality, thus they cannot be algebraic. In practice, however, this isn't a very useful method: "Almost all" transcendental numbers also obey the Thue–Siegel–Roth theorem.