  Given an unknown state of a qudit that has already been measured optimally,
can one still extract any information about the original unknown state?
Clearly, after a maximally informative measurement, the state of the system
`collapses' into a post-measurement state from which the {\em{same}} observer
cannot obtain further information about the original state of the system.
However, the system still encodes a significant amount of information about the
original preparation for a second observer who is unaware of the actions of the
first one. We study how a series of independent observers can obtain, or
scavenge, information about the unknown state of a system (quantified by the
fidelity) when they sequentially measure it. We give closed-form expressions
for the estimation fidelity, when one or several qudits are available to carry
information about the single-qudit state, and study the `classical' limit when
an arbitrarily large number of observers can obtain (nearly) complete
information on the system. In addition to the case where all observers perform
most informative measurements we study the scenario where a finite number of
observers estimate the state with equal fidelity,regardless of their position
in the measurement sequence; and the scenario where all observers use identical
measurement apparata (up to a mutually unknown orientation) chosen so that a
particular observer's estimation fidelity is maximized.
