Here is a solution with 8 fifth powers:

[5206, 2436, 432, 152, 69, 16, 7, 6, 5]

I'm not sure if this algorithm might miss a better solution

goal = 7**22

def strip(guess,n):
    """ start with first number, successive subtraction of guess^5 and
    count the number of 5th powers to sum to goal   """
    count = 0
    runningsum = guess**5
    values = [guess]
    diff = goal - runningsum
    while diff > 0:
        x = int(diff ** .2)
        x5 = x**5
        values.append(x)
        runningsum += x5
        diff = goal - runningsum
        count += 1
    return [count, values]

bestcount = 1000
bestvals = []
for n in range(5229,2,-1):
    info = strip(n,goal)
    thiscount = info[0]
    thesevals = info[1]
    if thiscount < bestcount:
        bestcount = thiscount
        bestvals = thesevals

print(bestcount)
print(bestvals)