What statistical tests should be performed to test effect of trials on response times where each subject has different number of data-points?
Mixed effects regression is somewhat related to repeated measures ANOVA and can handle missing data. However, they assume the missing data are missing at random.

Depending on what type of rejection criteria you are using, that assumption may be broken, in which case I think you're pretty much out of luck without a method to account for that. It may be that it is a sufficiently reasonable assumption, however, even if not strictly warranted.