Dropping The Lowest Grade

[the_geneticist]

For formative assessments (quizzes, weekly updates, reading summaries, etc), I think it's very reasonable to drop the lowest score or two.  Why?  Because formative assessments are a way to learn how much/how well you understand something.  Sometimes the answer is "not well at all".  And that's OK because that's part of learning something new.
For summative assessments (exams, projects, presentations), my vote is "NO".  I can't justify not holding students accountable for understanding a big chunk of course material.  I really dislike the "you don't have to take the final if you like your grade" UNLESS there was an exam that already tested their understanding of the most recent materials.  Otherwise, a savvy student could just skip the last 2-4 weeks of class.

Clarification: I understand your point and agree. In my system the final exam is a cumulative exam, they have already been tested on all of the material in the exams during the semester. The final is harder and longer than the regular exams, but is a second chance for students (can drop one of the semester grades and substitute the final). Thus, they still need to learn all of the material. But, since they need a "B" to pass the class, if they have a bad day on an exam, at least this gives them another shot.

Ah, that is a good strategy.  They still have to take the final exam and can show improvement from a previous exam. 
I've worked in departments where the faculty were trying to find ways to not have to give a final exam at all, even though they were technically required to give one.  The "if you like your grade, the final is optional" was a common strategy.