Example 1
The hungry hammerhead shark chased Alex all the way to the beach.
Duh-nuh… Duh-nuh… In this toothy example, the simple subject is shark. The modifiers—the, hungry, and hammerhead—are not part of the simple subject.
Example 2
Edward James Olmos autographed my Battlestar Galactica poster at the convention.
That's freaking awesome. Edward James Olmos is the man, and he's also the simple subject of this sentence. When the subject of a sentence is a proper noun, like it is here, the whole name is the simple subject.
Example 3
You can't forget your backpack in my car.
As this example shows, in an imperative sentence, the simple subject is you. Yes, you. Keep an eye on your Jansport!