hmmm...not certain what you mean...the value of Pi has been calculated to over a trillion digits.
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There is no biggest number. Numbers go on forever. Infinity isn't a number.
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.
The biggest number that can be possibly be written, is 10^4*185
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.