The history event designator !! is replaced by the last command in your history. Bash first prints out the command how it will be executed, then executes it.
Example:
$ foo
foo: command not found
$ !!
foo # command to be executed
foo: command not found # result of execution
In your case:
$ echo !
!
$ echo !!
echo echo ! # command to be executed
echo ! # result of execution
$ echo !!!
echo echo echo !! # command to be executed
echo echo !! # result of execution
Note that a command with an event designator is not inserted into history as is. It gets expanded and then entered to the history. That is why in the third command (echo !!!), the event designator is not replaced by echo !!, but by echo echo !.
Here is the last command again with the replaced part highlighted:
$ echo (!!)!
echo (echo echo !)! # command to be executed
echo echo !! # result of execution