How do I avoid a useless return in a Java method?

Question

The compiler's heuristics will never let you omit the last return. If you're sure it'll never be reached, I'd replace it with a throw to make the situation clear.

private static int oneRun(int range) {
    int[] rInt = new int[range+1]; // Stores the past sequence of ints.
    rInt[0] = generator.nextInt(range); // Inital random number.

    for (int count = 1; count <= range; count++) {
        ...
    }

    throw new AssertionError("unreachable code reached");
}

As @BoristheSpider pointed out you can make sure the second return statement is semantically unreachable:

private static int oneRun(int range) {
    int[] rInt = new int[range+1]; // Stores the past sequence of ints.
    int count = 0;

    while (true) {
        rInt[count] = generator.nextInt(range); // Add randint to current iteration.
        for (int i = 0; i < count; i++) { // Check for past occurence and return if found.
            if (rInt[i] == rInt[count]) {
                return count;
            }
        }
        count++;
    }
}

Compiles & runs fine. And if you ever get an ArrayIndexOutOfBoundsException you'll know the implementation was semantically wrong, without having to explicitly throw anything.

Since you asked about breaking out of two for loops, you can use a label to do that (see the example below):

private static int oneRun(int range) {
    int returnValue=-1;

    int[] rInt = new int[range+1]; // Stores the past sequence of ints.
    rInt[0] = generator.nextInt(range); // Inital random number.

    OUTER: for (int count = 1; count <= range; count++) { // Run until return.
        rInt[count] = generator.nextInt(range); // Add randint to current iteration.   
        for (int i = 0; i < count; i++) { // Check for past occurence and return if found.
            if (rInt[i] == rInt[count]) {
                returnValue = count;
                break OUTER;
            }
        }
    }
    return returnValue;
}

While an assert is a good fast solution. In general this kind of problems means that your code is too complicated. When I am looking at your code, it's obvious that you don't really want an array to hold previous numbers. You want a Set:

Set<Integer> previous = new HashSet<Integer>();

int randomInt = generator.nextInt(range);
previous.add(randomInt);

for (int count = 1; count <= range; count++) {
    randomInt = generator.nextInt(range);
    if (previous.contains(randomInt)) {
       break;
    }

    previous.add(randomInt);
}

return previous.size();

Now note that what we are returning is actually the size of the set. The code complexity has decreased from quadratic to linear and it is immediately more readable.

Now we can realize that we don't even need that count index:

Set<Integer> previous = new HashSet<Integer>();

int randomInt = generator.nextInt(range);

while (!previous.contains(randomInt)) {          
    previous.add(randomInt);      
    randomInt = generator.nextInt(range);
}

return previous.size();