  We develop a nonparametric extension of the sequential generalized likelihood
ratio (GLR) test and corresponding time-uniform confidence sequences for the
mean of a univariate distribution. By utilizing a geometric interpretation of
the GLR statistic, we derive a simple analytic upper bound on the probability
that it exceeds any prespecified boundary; these are intractable to approximate
via simulations due to infinite horizon of the tests and the composite
nonparametric nulls under consideration. Using time-uniform boundary-crossing
inequalities, we carry out a unified nonasymptotic analysis of expected sample
sizes of one-sided and open-ended tests over nonparametric classes of
distributions (including sub-Gaussian, sub-exponential, sub-gamma, and
exponential families). Finally, we present a flexible and practical method to
construct time-uniform confidence sequences that are easily tunable to be
uniformly close to the pointwise Chernoff bound over any target time interval.
