There isn't one.
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Since there is no biggest integer, there can be no biggest rational number - since any integer is a rational number with the denominator = 1.
The biggest integer "between" -9 and -3 is -4.
-1 is the largest negative integer.
A mixed number consists of an integer followed by a proper fraction. 6 is the biggest integer, .5 remains. .5 6 1/2
There is no such number. According to Peano's axiom, each positive integer has a successor, which is larger (one more) than it. So the set of integers goes on for ever.