  We analyze almost sure asymptotic behavior of extreme values of a
regenerative process. We show that under certain conditions a properly centered
and normalized running maximum of a regenerative process satisfies a law of the
iterated logarithm for the $\limsup$ and a law of the triple logarithm for the
$\liminf$. This complements a previously known result of Glasserman and Kou
[Ann. Appl. Probab. 5(2) (1995), 424--445]. We apply our results to several
queuing systems and a birth and death process.
