  The notion of a non-perturbative effect is ambiguous if it requires the
subtraction of a perturbative part defined by a diverging series. A common
procedure consists in dropping the order of minimal contribution and the higher
orders. This allows us to isolate very accurately the one-instanton effect for
the double-well potential. For the one plaquette gauge theory, an exact
analytical expression can be written for the non-perturbative part. We report
recent attempts to extend this approach to the average plaquette of quenched
QCD. Our goal is to express the non-perturbative effects in terms of
expressions of the form beta^B exp(-A beta) calculable semi-classically. The
situation is complicated by zeroes of the partition function in the complex
$\beta$ plane (presumably near 5.75 pm i 0.2). We discuss two methods to
describe the intermediate and large order behavior of the perturbative series.
One is inspired by mean field theory (logarithmic specific heat) and reproduces
accurately the known perturbative series with only two free parameters. A
diagrammatic interpretation of this fact is still lacking. The other is based
on infra-red renormalons with a factorial growth showing up at order larger
than 20 and a possible effective theory interpretation. These extrapolations
are compatible with the non-perturbative part of the plaquette being
proportional to a^4. We propose an exponential parametrization to the
corrections to the universal part of the beta function and find results
compatible with the suggestion of a^2 corrections made by C. Allton.
