  The maximum type-I and type-II error exponents associated with the newly
introduced almost-fixed-length hypothesis testing is characterized. In this
class of tests, the decision-maker declares the true hypothesis almost always
after collecting a fixed number of samples $n$; however in very rare cases with
exponentially small probability the decision maker is allowed to collect
another set of samples (no more than polynomial in $n$). This class of
hypothesis tests are shown to bridge the gap between the classical hypothesis
testing with a fixed sample size and the sequential hypothesis testing, and
improve the trade-off between type-I and type-II error exponents.
