It seems to me the time complexity of this is non-linear in the amount as you can see in the way the number of iterations required grows. Unless maybe I've messed something up in my translation from Clojure to Python.

jake-change.py

#!/usr/bin/env python3

def count_ways_to_make_change(coins, total):

    iterations = 0

    ways       = 0
    digit      = 0
    counts     = [0] * len(coins)

    while digit < len(counts):
        iterations += 1

        counts[digit] += 1
        new_sum = sum(d * c for d, c in zip(coins, counts))

        if new_sum < total:
            digit = 0
        else:
            if new_sum == total: ways += 1
            counts[digit] = 0
            digit += 1

    return ways, iterations


if __name__ == '__main__':

    coins = [ 1, 5, 10, 25 ]

    for amount in [ 100, 200, 400, 800 ]:
        ways, iterations = count_ways_to_make_change(coins, amount)
        print('amount: {}; ways: {}; iterations: {}'.format(amount, ways, iterations))