When I make pancakes, the recipe I use calls for 1 egg and 1c of flour. However, usually I increase the flour (up to 50%) to make a bit more. I've tried both keeping 1 egg, or adding a second one, but it doesn't seem to make any difference...in other words, it seems like there's an awful lot of room to maneuver when it comes to how much egg to add. If so, I could add 3 or 4 eggs to get more protein into the pancakes. How would adding lots of extra eggs affect the end result?
This answer assumes the US-type of pancake.
There are some pancake recipes with no egg at all. So the question is, what does the egg do in a pancake recipe?
It contributes soem fat and emulsifiers from the yolk, some water, and a bunch of protein (mostly from the whites).
This will add structure to the pancake, some eggy flavor and richness, and perhaps somewhat mitigate the danger of toughening the pancake through over mixing, by inhibiting gluten formation (mostly due to the fat and lecithin from the yolk).
However, as you have already discovered, the egg is not completely essential to the pancake, which basically a griddled quick bread, and can easily get all of the structure it needs from the gelatinization of the starch from the flour.
Adding lots more eggs would start to move you into custard or quiche-like territory, where the product would have a lot more egg flavor, and start to take on a more scrambled-egg type texture depending on how far you go. You will get less flavor but more textural change from using just whites.
If you love pancakes, I would make them with any pancake recipe you like, and enhance your protein intake from other food items. Personally, I love a scrambled egg, and they go great with pancakes.
Edit: I see in your profile that you are in France. Since we also call French style crepes the same thing in English, I am thinking you probably do mean US style pancakes, so the above would apply.
If you are thinking of crepes, they tend to be a touch eggier than pancakes to start with, which helps give them the structure to role or fold easily while still being so thin.
I imagine that they could take a bit more egg still, with some change in texture and more egg flavor, but might start to stick to the pan more, as egg really likes to stick. At the extreme end, you would essentially have thin fried scrambled egg, which you can roll up with fillings, as some other cuisines do.
Still, I am not an expert on crepes; I have never made them myself. If that is what you are asking about, hopefully someone else can give you a more knowledgable answer.
If you are using an American US-style pancake recipe (or any American recipe for that matter) in the EU a point to remember is that EU egg sizes are bigger for their number than US sizes. An EU large egg is bigger than an American, an EU Medium egg is the equivalent of a US Large egg. So if you are using an american recipe that uses large eggs use medium eggs in the EU if you are trying to be precise. Conversely, use US XL eggs if using a European recipe.
The upshot for @Kelsey is that using an EU large egg on an american recipe but adding more flour probably gets the recipe right from a proportion standpoint.
Here is the wikipedia egg reference page
I put 3 eggs in a two egg pancake recipe all the time and notice absolutely no difference at all.