Sure, I can help you with the longest_run() function. Here is one possible implementation:
```
def longest_run(outcomes):
    max_run = 1
    curr_run = 1
    for i in range(1, len(outcomes)):
        if outcomes[i] == outcomes[i-1]:
            curr_run += 1
            if curr_run > max_run:
                max_run = curr_run
        else:
            curr_run = 1
    return max_run 
``` 
This function takes in an array of dice rolls and returns the length of the longest run of consecutive rolls with the same value. It works by iterating through the array and keeping track of the current run and the maximum run seen so far. If the current roll is the same as the previous roll, the current run is incremented. If the current run is longer than the maximum run, the maximum run is updated. If the current roll is different from the previous roll, the current run is reset to 1.

You can use this function in your run_experiments() function to calculate the length of the longest run for each experiment:
``` 
def run_experiments(sides = 6, rolls = 100, num_experiments = 1000):
    all_max_runs = []
    for _ in range(num_experiments):
        outcomes = roll_die(sides = sides, rolls = rolls)
        max_run = longest_run(outcomes)
        all_max_runs.append(max_run)
    mean_max_run = np.mean(all_max_runs)
    return mean_max_run
``` 
Hope this helps!
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