We are tasked with finding the units digit of the sum $1^1 + 2^2 + 3^3 + \dots + 2020^{2020}$.

To solve this, we need to observe the behavior of the units digits for powers of numbers and recognize patterns. Let’s break the process down.

Step 1: Understanding Powers and Units Digits

Each number’s powers repeat in cycles for their units digits. Let's go over some examples to show how this works:

• 1^n: The units digit is always 1 for any power of 1.
• 2^n: The units digit follows a cycle: 2, 4, 8, 6, and repeats every 4 terms.
• 3^n: The units digit follows a cycle: 3, 9, 7, 1, and repeats every 4 terms.
• 4^n: The units digit alternates between 4 and 6 (cycle length 2).
• 5^n: The units digit is always 5 for any power of 5.
• 6^n