  Speedable numbers are real numbers which are algorithmically approximable
from below and whose approximations can be accelerated nonuniformly. We begin
this article by answering a question of Barmpalias by separating a strict
subclass that we will refer to as superspeedable from the speedable numbers;
for elements of this subclass, acceleration is possible uniformly and to an
even higher degree. This new type of benign left-approximations of numbers then
integrates itself into a hierarchy of other such notions studied in a growing
body of recent work. We add a new perspective to this study by juxtaposing this
hierachy with the well-studied hierachy of algorithmic randomness notions.
