  It is known that, after formation, a Rydberg wave packet undergoes a series
of collapses and revivals within a time period called the revival time, $t_{\rm
rev}$, at the end of which it is close to its original shape. We study the
behavior of Rydberg wave packets on time scales much greater than $t_{\rm
rev}$. We show that after a few revival cycles the wave packet ceases to reform
at multiples of the revival time. Instead, a new series of collapses and
revivals commences, culminating after a time period $t_{\rm sr} \gg t_{\rm
rev}$ with the formation of a wave packet that more closely resembles the
initial packet than does the full revival at time $t_{\rm rev}$. Furthermore,
at times that are rational fractions of $t_{\rm sr}$, the square of the
autocorrelation function exhibits large peaks with periodicities that can be
expressed as fractions of the revival time $t_{\rm rev}$. These periodicities
indicate a new type of fractional revival occurring for times much greater than
$t_{\rm rev}$. A theoretical explanation of these effects is outlined.
