  The Aschenbach effect, the increasing behavior of the angular velocity of a
timelike circular orbit with its radius coordinate, is found to extensively
exist in rapidly spinning black holes to a zero-angular-momentum observer. It
also has potential observation in the high-frequency quasi-periodic
oscillations of X-ray flux. However, observing such effect remains to be a
challenge in static and spherically symmetric black hole backgrounds. In this
paper, we mainly focus on such issue. Starting with the geodesics, we
analytically study the underlying properties of the timelike circular orbits,
and show the conditions under which the Aschenbach effect survives. We then
apply it to three characteristic black holes exhibiting different features. The
results state that this effect is absent for both the Schwarzschild and
Reissner-Nordstr\"{o}m black holes. While, for the dyonic black hole in
quasi-topological electromagnetics, there indeed exists the Aschenbach effect.
This provides a first example that such effect exists in a non-spinning black
hole background. Moreover, it also uncovers an intriguing property for
understanding the black holes in nonlinear electrodynamics.
