These rats needs a game to go with them! They're so cute.
I never managed to get to the "Pope Rat", because the exponential scaling was a bit much to take. This is a type of scaling that almost never works, in any game, except if you really delve into it and explore different sense of scales (Universal Paperclip, Cookieclicker) or use it as the sole mechanic (2048).
Without any deviation to that "doubling", you leave no room for player expectation. If the player knows that the next 2 minutes is going to be exactly like the last 2 minutes, just with even more clicks, it's hard to keep them motivated. They need something to optimize or some agency in the direction the game takes, instead of feeling like they're only making the frames go by.
Examples of what you could have done to achieve this is variation to the "tech tree" of rats, like having to combine varying number of certain kinds of rat. Think "Little Alchemist", simple and addictive to find combinations.
Since you also had the ability to pick up and relocate rats, a Lemmings style mechanic would be right at home. You could have the rats walk around and getting into places they shouldn't, like in the combinator when you're trying to combine two other rats, so you need to drag them away. A hamster wheel with limited space to place them in, for example. The player would have to need to balance how many rats / how productive / how fast they want to progress against how many rats they can keep track of.