Say I have a set of n measurements. The measurement process has a known error. I can't assume that the true values being measured follow a normal distribution. How can I estimate the actual maximum or minimum value of the set given this information (accounting for the fact that, as n increases, the max will become larger and min smaller even if the true values are all identical)?
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In your case, I would turn to the nonparametric Komolgorov-Smirnov one-sample test. With this test you compare a set of measurements against a known cumulative distribution function. The expression to test is
$ \begin{split} D=\max_{i \; \in \; \{1 \ldots N\}}(\mid F_o(X_i) - S_N(X_i) \mid) \end{split} $
with $F_o(X)$ a completely specified cumulative distribution function and $S_N(X)$ the observed cumulative distribution function. As can be seen, the test focuses on maximal deviations in the absolute sense. When $D$ deviates more than expected then the test rejects $H_0$ because of the maximal absolute deviation in the sample.
You can use any of known theoretical cumulative distribution function, which is why this test is nonparametric.
