Rocky is a rock which Patrick mistook for a snail. Patrick entered Rocky in a snail race competing against Gary and Squidward's snail, Snellie. This rock actually won the Snail Race somehow like a snail with a rock for a shell in the episode The Great Snail Race.
Rocky the "rock" is brown in color and is the same size as a snail (Gary, Snellie, and Larry Luciano) Also, it looks like a scaled down version of Patrick's rock.
In Buried in Time, Patrick wants to donate Rocky to the Time Capsule. In this episode, however, Rocky is lumpy and gray, maybe meaning that he got older.
In The Great Snail Race, Rocky did not move but at the end he was last seen on the finish line.
Perhaps Mayor put Rocky at the finish line.
Buried in Time is Rocky's second appearance
According to Patrick, Rocky has nerves of steel. Patrick even kept saying "I know, I know, he's got nerves of steel."
In Buried in Time, Rocky looks lumpier and he is gray.
A smaller, bluish-gray version of Rocky (with a face) is seen in Sentimental Sponge. However, it is known as Pete.
Patrick could actually have three pet rocks and two are called Rocky and one is named Pete.
Rocky was also in the beginning of Patrick's Staycation. When SpongeBob said "Ooh, a Shadow Puppet Show" Patrick said "SpongeBob Please." While holding a rock that looked like his own.