finding the largest factorial that is less than the number entered

The while loop doesn’t have curly braces, so the only thing in the loop’s body is the for loop. num starts at 1, and nothing inside the loop ever increases it, so it would loop forever.

You don’t need a nested loop, though – a single loop that calculates the factorial as you go should suffice:

int numinput = 150; //number than we are trying to find the largest factorial less than
int num = 1;
int factorial = 1;

while (factorial < numinput) {
    num++;
    factorial *= num;
}

// We overshot to terminate the loop, go back one number
factorial /= num;
num--;

System.out.println
    ("The largest factorial less than " + numinput + " is " + num + "!, or " + factorial);