Big-O isn't as relevant as it once was. Modern hardware includes mutithreading, piplining, numa, and complex caching. Some operations can take less than one CPU cycle and others can take hundreds, or exceptionally thousands of cycles. Trying to describe the behavior of algorithms solely as a function of the number of "trips through the innermost loop" can be a very misleading description!
Besides that, other measures such as big-Omega should be referenced in any discussion of big-O.
(I did enjoy Big-O the anime series though! /grin)
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