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.