If you ground enough beans for 4 cups of coffee but only brew one cup of coffee how much stronger will it be?
My gut feel is the first initial pass of water will pull more coffee materials from the grounds and making a stronger nicer cup, but I have no clue if that impression is right.
It will be close to four times as strong. I say "close to" because the amount of coffee can influence some other things.
How long the water is in contact with the grounds affects the strength. Depending on the brew method, a significantly deeper bed of grounds can slow down how fast water passes through (if water takes longer to get through, it will be stronger). The finer the grind, the more the effect. The biggest impact would probably be with a cone-shaped filter.
If you're grinding it yourself rather than using coffee pre-ground for a filter brewer (standard grind for store-bought ground coffee), using too fine a grind can slow down the water so much that the filter basket can overflow (and you can get weaker coffee because much of the water bypasses the grounds).
Note that if you increase the brew time significantly, the finished coffee can have a different flavor profile (generally not as good). The coffee can get over-extracted, in which case flavor compounds that tend to be more bitter and sour become more noticeable.
The grounds retain some of the water, roughly 2 grams of water for every gram of coffee (a paper filter also retains some water). If you measure using a coffee scoop, coffee makers typically come with a 1 tablespoon size scoop, which will hold somewhere around 5 to 7.5 grams of coffee depending on the coffee, grind size, and whether you level the scoop or use a rounded scoop. So every 2-3 scoops of coffee with retain about 1 fl oz of water. If I brew a 12 oz mug of coffee, I need to start with about 14 oz of water to get 12 oz of coffee out.
If you start with enough grounds for four cups of coffee and brew just one cup, the coffee will retain a significant amount of water relative to your serving. The relationship between the coffee:water ratio and extraction is based on all the water and all the coffee rather than the water in the brew that ends up in your cup.
So you need to include the retained water when figuring the effect on brew strength. Enough grounds for four 6 oz cups (the serving size drip coffee makers are based on), at standard strength, will retain roughly 3 oz of water. So if you don't account for the retained water, you'd end up with a serving of about 3 oz. You would need to start with water at the 1 1/2 cup level to end up with a one cup serving. If you were making four cups, the volume loss from retained water wouldn't be that noticeable since cups normally aren't filled to the brim.
The strength of what is added to the brew does vary from the beginning to the end of the extraction, but that shouldn't be a significant factor in brew strength for the batch in this case.
