It looks like magic at first how keyword parameter with list value keeps its value across calls. Here is version of code which use 'function as object' and stores value as attribute of the function, and usage example (practically maybe it would be safer the code to be included in code itself and not used as module, but principle is same, and this is yet intrinsically unsafe as passwords are visible in source code.

try:
    import tkinter as tk
except ImportError:
    import Tkinter as tk
# from http://effbot.org/tkinterbook/entry.htm

failure_max = 3
passwords = [('DaniWeb', 'best1'), ('newbie', 'help!help!')]

def make_entry(parent, caption, width=None, **options):
    tk.Label(parent, text=caption).pack(side=tk.TOP)
    entry = tk.Entry(parent, **options)
    if width:
        entry.config(width=width)
    entry.pack(side=tk.TOP, padx=10, fill=tk.BOTH)
    return entry

def enter(event):
    check_password()

def check_password():
    """ Collect 1's for every failure and quit program in case of failure_max failures """
    #print(user.get(), password.get())
    if (user.get(), password.get()) in passwords:
        check_password.user = user.get()
        root.destroy()
        print('Logged in')
        return
    check_password.failures += 1
    if check_password.failures == failure_max:
        root.destroy()
        raise SystemExit('Unauthorized login attempt')
    else:
        root.title('Try again. Attempt %i/%i' % (check_password.failures + 1, failure_max))
check_password.failures = 0

root = tk.Tk()
root.geometry('300x160')
root.title('Enter your information')
#frame for window margin
parent = tk.Frame(root, padx=10, pady=10)
parent.pack(fill=tk.BOTH, expand=True)
#entrys with not shown text
user = make_entry(parent, "User name:", 16, show='*')
password = make_entry(parent, "Password:", 16, show="*")
#button to attempt to login
b = tk.Button(parent, borderwidth=4, text="Login", width=10, pady=8, command=check_password)
b.pack(side=tk.BOTTOM)
password.bind('<Return>', enter)

user.focus_set()

parent.mainloop()

Usage assuming that above code can be found by import as tkPasswd

import time

try:
    import tk_passwd
except SystemExit as e:
    print(e)
else:   
    user = tk_passwd.check_password.user
    print('Welcome to IRS Tax Database, %s!' % user)
    time.sleep(4)
    print('Just kidding!')

You can choose your way but knowing username is making breaking easier. For example internet service of my bank is hiding both user code and single use pin number.