How to store results from while loop and sentinel in python?

You don't want to have a fixed number of quizes or projects. Instead, use a loop for each of those types of scores, so you can keep asking until they user doesn't have any more scores to enter.

I'm not going to write the whole thing for you, but here's one way to handle the quizes:

quiz_scores = []

while True:
    score = int(input("Quiz #{} ----- ".format(len(quiz_scores)+1)))
    if score == -1:
        break
    quiz_scores.append(score)

quiz_total = sum(quiz_scores) - min(quiz_scores) # add up the scores, dropping the smallest

There are other ways you could do it. For instance, instead of building a list of scores, you could keep track of a running sum that you update in the loop. You'd also want to keep track of the smallest score you've seen so far, so that you could subtract the lowest score from the sum at the end.

If you're storing the input as integers, you probably want a dict. You'd do something like this:

numberOfInputs = 10
names = ["Quiz 1", "Quiz 2", "Program 1", "Exam 1", "Final Exam"]
d = {}
for name in names:
    i = int(input(name.rjust(4)))
    if i < 0:
        quit()
    d[name] = i
# calculate the ave
for name in d:
    print(name)
print("Average: " + ave)

You'd check if you need to quit every time and you'd also have a fairly intuitive way of accessing the data.

I used some things that might come in handy in the future, if you continue to program in python (like generators, list comprehension, map)
You can try something like this: (Can be simplified, but for clarity.. )

#!/bin/env python3

"""
Course grade computation
"""

# quiz.py

import sys


def take(name="", dtype='int', sentinel=-1):
    """:param dtype: string representing given input type"""
    counter = 0
    i = 0
    while True:
        i = eval(dtype)(input("{} #{}: ".format(name,counter)))  # We should use function map instead, but for simplicity...
        if i > 10 or i < -1:
            print("Invalid input: input must be in range(0,10) or -1", file=sys.stderr)
            continue
        if i == sentinel:  # <-- -1 will not be added to list
            break
        yield i
        counter += 1


def main():
    grades_map = dict([(100, 'A'), (90, 'B'), (80, 'C'), (70, 'D'), (60, 'F')])

    print("Input quiz scores")
    quizes = [i for i in take("Quiz", dtype='float')]
    quiz_sum = sum(quizes)
    quiz_lowest = min(quizes)

    print("Input project scores")
    projects = [i for i in take("Project", dtype='float')]
    proj_sum = sum(projects)

    print("Input exam scores")
    exam1, exam2, final = map(float, [input("Exam #1:"), input("Exam #2:"), input("Final:")])

    total = ...  # Didn't know what the total really is here
    average = ...  # And same here

    grade = grades_map[round(average, -1)]

    # Handle your prints as you wish


if __name__ == '__main__':
    main()

EDIT: Change in the generator so that -1 is not added to the list