It's probably hard to write a deep game about doing naught more than repeatedly clicking a big red button you have been instructed not to click, but it shouldn't be impossible to write a funny one. Big Red Button's problem is that it's just not witty. The game speaks to you in a harassed tone each time you click the button in defiance of the instruction at the top of the screen, and what it says is sloppy and inconsistent, and hasn't been proofread. It's like the first draft of a comedy sketch that doesn't have a direction or any quality yet, just a basic idea. As such, it's unable to say anything about the one-move games it's probably trying to mock. Worse crimes are that it didn't make me laugh and it doesn't even understand PRESS in place of CLICK.
The game comes in two flavours. The everlasting one loops its messages forever in response to your clicking of the button. The non-everlasting one ends. Both games contain the same messages, and it's unlikely anyone would want to read them more than once, so the difference makes little difference.