How many times can a Prem side face each other in a single season?

First of all, what competitions will both Chelsea and Manchester City play in? (Well, I just use them as an example really to make things easy, any Premier League teams will do the trick hypothetically.)

Say if both sides finished top 4 in the previous season, both teams will qualify for the Champions League. Obviously, they will still (Will Still goat) play in the Premier League. Also, as Premier League sides they automatically qualify for the FA Cup and League Cup. Therefore, it looks like the most times that Chelsea and City play each other is 4.

I'm sorry to tell you things is not that easy. You see in Premier League, teams play each other TWICE. In the Champions League, unless it's the final, teams play each other TWICE also. Although an FA Cup is usually played one game, a tie will lead to a replay, which means Chelsea and City can play each other TWICE in the competition. Finally, a two-leg tie is played when the League Cup reaches the semi-final, which means...

IN FOUR COMPETITIONS, CHELSEA AND MANCHESTER CITY CAN PLAY EACH OTHER UP TO 8 TIMES!

Well, that's the end- Not too soon! Things started to get a little bit complicated for now on. The winner of FA Cup will play the League winner in a match called Community Shield. Say if Chelsea won the League and City won the FA Cup, they will face each other in the McDonald's Shield (coz it's sponsored by McDonald's). The total match-up is now 9. But wait a minute! What if, very unluckily, Chelsea and City have the same points, same goal difference, same goals scored and the same amount of games won, drawn and lost in the league? An additional play-off in required to decide to League winner, because surely you won't want to see both teams holding hands on the trophy presentation ceremony, right? (I'm not saying it hasn't happened before, but things are different nowadays.) Thus, we can see this banging clash between Chelsea and City up to 10 times possibly!

Now I'm not a huge fan of Gianni Infantino, but the reason why I do the Club World Cup separately is that... I think it would be a paragraph to big if I stuff this all in. The FIFA Club World Cup is basically 6 winners from 6 different continents (as far as I know there ain't an Antarctica FC) and the winner of the highest tier league of the host country play against each other for a mickey mouse cup. Say if City went on to win the Champions League (nobody except City fans will want to see this) and Super Cup, they will be representing Europe to play in the CWC. Chelsea, on the other hand, will play the competition as the Premier League winner. So, theoretically Chelsea can face Man City 11 times in a season (and beat their ass 11 times) if everything goes as planned.

With all of that said, you may still think there are more than 11 times that Chelsea and City (or any two Premier League sides) can face other. Feel free to comment and I'll continue my research on this topic.