  We introduce a magnetically charged extremal regular black hole in the
coupled system of Einstein gravity and nonlinear electrodynamics. Its near
horizon geometry is given by $AdS_2\times S^2$. It turns out that the entropy
function approach does not automatically lead to a correct entropy of the
Bekenstein-Hawking entropy. This contrasts to the case of the extremal
Reissner-Norstr\"om black hole in the Einstein-Maxwell theory. We conclude that
the entropy function approach does not work for a magnetically charged extremal
regular black hole without singularity, because of the nonlinearity of the
entropy function.
