Try using this code:
#Simple Lotto program
import random
import time
picked = 0
#Greeting
print('''Hello! Welcome to Xog\'s Lotto Application. Thank you for taking the time to test my application out.
To play this Lotto game, I want you to pick three numbers from 1 through 5.''')
#Waits for 2 seconds.
time.sleep(2)
#Getting to know you
print('Don\'t get hasty! First, I want you to tell me your name. Please type it in and press Enter.')
name = raw_input()
print('Hello, ' + name + ', my name is Xog.')
#Number picking
number = random.randint(1, 5)
print('Okay, now that we know eachother a little bit, I think it\'s time we begin.')
while picked < 3:
print('Please pick a number from 1 through 5.')
playerPicked1 = raw_input()
playerPicked1 = int(playerPicked1)
picked = picked + 1
print('Pick your second number, between 1 and 5, that you did not pick.')
playerPicked2 = raw_input()
playerPicked2 = int(playerPicked2)
picked = picked + 1
print('Pick your third and last number, between 1 and 5, that you did not pick.')
playerPicked3 = raw_input()
playerPicked3 = int(playerPicked3)
picked = picked + 1
#Your issue was here. You can't do playerPicked1 or playerPicked2 or playerPicked3 == number.
if (playerPicked1 == number) or (playerPicked2 == number) or (playerPicked3 == number):
number = str(number)
print('The lotto number was ' + number + '! Congratulations! You\'ve just won $500 Hyperspace Dollars!')
time.sleep(3)
print('Well.. Okay, maybe not the money.')
else:
number = str(number)
print('Sorry, the last number was ' + number + '.')
print('Thanks for playing Xog\'s lotto!')
What I saw wrong was first of all, you used input(). Use raw_input(). input() expects a valid python expression for input. Also the reason why this wasn't working at all was you did playerPicked1 or playerPicked2 or playerPicked3 == number. It should be (playerPicked1 == number) or (playerPicked2 == number) or (playerPicked3 == number). And instead of making a new if statement you could have used an else.
