Update: I have solved the problem of compounding factorials by resetting the factorial to 1 at the start of the loop and eliminating unnecessary code. Here is the new version.
more = "yes"
while more == "yes":
n = int(input("Enter a number to get its factorial: "))
if n >= 0:
factorial = 1
for num in range(1, n+1):
factorial *= num
print(n,'!=', factorial)
more = input("Would you like to get another number's factorial?:")
else:
print("Factorials are not defined for negative integers.")
print("Thank You!")
You need to reset the factorial variable to 1 at the start of the while loop.
Also, you'll want to check if n is positive for every input, as it currently only does so for the first.