When Joe is talking to Cid in the kitchen at night, Cid has a noticeable gap in one of his front teeth. In the next scene Cid is talking to his mother Sarah and still has a very wobbly baby tooth where the gap was previously.
When the mob is preparing to send back Old Joe, there are four gangsters. Two restraining him, one attaching the gold bars and another preparing the time machine. In the fight that follows, there are only three of them.
When the truck is upended and overturns, it is clear that there is no one in the cab. Then when it comes to rest upside down on the road, the close-up reveals the Sara and Cid characters held inside by their seat belts.
When Joe and Old Joe are in the café together, the waitress Beatrix comes over to take their order, and their table is empty. However when she returns to give them their food, there are two drinks already on the table. We know that she has not given them their drinks order previously because the rest of the scene is a conversation between the 2 characters in which she does not reappear.
When Cid is answering multiplication questions with Sara and answers the 7x3 question for the 21 tile, which is then placed on the board, it disappears in the next shot, reappears in the following shot and disappears again in the next shot.
When Joe and Cid are having a conversation looking out of the hatch at the end of the tunnel from the house, the branch holding up the hatch changes position from shot to shot. It is rotated about 90 degrees between the positions and the position changes four or five times during the conversation.
Before Joe's first meeting with Abe, Kid Blue is messing around with his gun, eventually winding up pointing it at Joe and cocking the hammer. The film cuts to a shot of Joe, then to a shot of Kid Blue pointing the gun at Joe with the hammer uncocked. The next shot of Kid Blue shows him with his thumb on the cocked hammer.
When Sarah is doing multiplication tables with Cid, the first tile we see him place (56) moves repeatedly between shots - it's correctly on 7x8 from his point of view, but the tile is on 7x9 when we move to her point of view. The next tile (32), which he supposedly places on 8x3, is actually placed on 7x8, adjacent to the incorrectly-placed first tile from Sarah's point of view.
When Joe and Seth are talking in Joe's apartment towards the beginning of the movie, the streams of tears on Seth's cheeks and upper lip change multiple times in between the close-up shots of his face.
When Young Joe closes his loop, a paradox is established.
Old Joe no longer exists, so he can't be sent back in time.
Cid never sees his 'mom' killed, so Cid doesn't become the Rainmaker.
Old Joe (who no longer exists) can't be a threat to the Rainmaker (who also doesn't exist), so there is no reason for Young Joe and Cid to ever meet, and no reason for Young Joe to kill himself.
Joseph Gordon-Levitt went through subtle prosthetic makeup around his nose, lips, and brow to look more like Bruce Willis, but the makeup artists failed to synchronize the actors' ears. Willis has lobed ears while Gordon-Levitt has attached ears.
Young Joe disposes of bodies, by throwing his victims downwards a shaft, ending up into the fire. The actual body we see ending up into that fire seems to slide horizontally towards the fire, instead of vertically falling into it, given the orientation of the fire.
When the younger version of Joe's older self is waiting for HIS older self to appear, stubble is shown clearly on his face. After he shoots his older self he bends down to collect his silver/gold bars and the stubble on his face is gone.
After Cid blows up Jesse at the farm and runs into the field, as Joe approaches him, his face is full of blood. Yet, in the blood, you can clearly see the outline of goggles or glasses that were used to protect his eyes.