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Rules of the game:
a) The goal is to find a /practical/ example, such as an
experiment that can be readily performed. In particular
this excludes Gedankenexperiments on cosmological scales
or ultra-subatomic scales where we can't actually do the
experiment. This most explicitly excludes objections to
the phrasing of the definition; if a definition is operationally
usable, it is good enough for me, regardless of phrasing.
b) There must be a reasonable way to analyze the experiment
in terms of energy.
c) The analysis must fail to describe some /observable/
outcome of the experiment, and the failure must be due to
the lack of a "good" definition of energy. In particular
this excludes fussing about gauge independence, since
the gauge is unobservable.
I've asked this question before, and received a grand total
of zero valid examples.
That's why I continue to believe that for all practical
purposes, energy is well defined.