My solution with the current year calculated for me

from datetime import date

# Step 1
# Ask the user for their name and the year they were born.
name = input("Please enter your name: ")

while True:
    birthyear = input("What year were you born? ")
    try:
        birthyear = int(birthyear)
    except ValueError:
        continue
    else:
        break

# Step 2
# Calculate and print the year they'll turn 25, 50, 75, and 100.
age25 = birthyear + 25
age50 = birthyear + 50
age75 = birthyear + 75
age100 = birthyear + 100

# Step 3
# If they're already past any of these ages, skip them.
currentYear = int(date.today().strftime("%Y"))
if age25 > currentYear:
    print ("{}, you will turn 25 in {}, 50 in {}, 75 in {}, and 100 in {}".format(name, age25, age50, age75, age100))
elif age50 > currentYear:
    print ("{}, you will turn 50 in {}, 75 in {}, and 100 in {}".format(name, age50, age75, age100))
elif age75 > currentYear:
    print ("{}, you will turn 75 in {}, and 100 in {}".format(name, age75, age100))
elif age100 > currentYear:
    print ("{}, you will turn 100 in {}".format(name, age100))
else:
    print ("{}... Damn! You're old!".format(name))