Joshua likes to play with numbers and patterns.  Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$.  Part of the reason Joshua likes the number $6$ so much is that the powers of $6$ all have the same units digit as they grow from $6^1$:
\begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6^4&=1296,\\6^5&=7776,\\6^6&=46656,\\\vdots\end{align*}
However, not all units digits remain constant when exponentiated in this way.  One day Joshua asks Michael if there are simple patterns for the units digits when each one-digit integer is exponentiated in the manner above.  Michael responds, "You tell me!"  Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers and see what he can discover.  "See if you can find the units digit of $2008^{2008}$," Michael challenges.  After a little while, Joshua finds an answer which Michael confirms is correct.  What is Joshua's correct answer (the units digit of $2008^{2008}$)?
To find the units digit of \(2008^{2008}\), we need to focus on the units digit of the base number, which is \(8\). We will determine the pattern of the units digits of powers of \(8\).

1. **Identify the units digit pattern of powers of \(8\):**
   \[
   \begin{align*}
   8^1 & = 8 \quad (\text{units digit is } 8) \\
   8^2 & = 64 \quad (\text{units digit is } 4) \\
   8^3 & = 512 \quad (\text{units digit is } 2) \\
   8^4 & = 4096 \quad (\text{units digit is } 6) \\
   8^5 & = 32768 \quad (\text{units digit is } 8) \\
   \end{align*}
   \]
   We observe that the units digits repeat every 4 powers: \(8, 4, 2, 6\).

2. **Determine the position of \(2008\) in the cycle:**
   Since the units digits repeat every 4 powers, we need to find the remainder when \(2008\) is divided by \(4\):
   \[
   2008 \mod 4 = 0
   \]
   This means \(2008\) is a multiple of \(4\), and thus, it corresponds to the 4th position in the cycle.

3. **Identify the units digit at the 4th position in the cycle:**
   From the pattern \(8, 4, 2, 6\), the 4th position has the units digit \(6\).

Therefore, the units digit of \(2008^{2008}\) is \(\boxed{6}\).