There does not exist a largest negative rational number. prove by contradiction

If there WERE a largest negative rational number, then you could
write it down on a piece of paper, and come running into my house
yelling "I've got it!  I've got it!  I know the largest negative rational
number!  HERE IT IS!"

I could calmly take the paper from you, unfold it, and look at it.
No matter how large your number is, I could subtract 1 from it.

Then I would have a negative rational number that's LARGER
than your number.

This contradicts my original assumption up there at the beginning
of my answer.  There is NO largest rational negative number.