Infinity (it goes on forever)
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The biggest number that can be possibly be written, is 10^4*185
Googolplex. However it is useless to science because it exceeds the number of particles in the universe!
The biggest named number. The entire universe would be too small to write this number down, and counting to it would take trillions of trillions of years.
Since there is no biggest integer, there can be no biggest rational number - since any integer is a rational number with the denominator = 1.
997 is the biggest 3 digit Prime number.