Overview

Here are the main parts of the game boy unit:

The second block diagram (from patent 5134391) shows that the LCD controller is split into two parts.

and here is the main PCB from a real game boy:

You can see the main CPU in the middle, the two SRAM chips  used for display and work memory and I think the fourth chip amplifies the sound. At the bottom you can see where the cartridge (game pak) connector has been removed.

In the block diagram "6-pin subconnector" is the link cable port, here are pinouts for it and the cartridge connector

I think that the CPU itself outputs a stream of pixels which the two LCD chips (see the schematics) then decode into vertical and horizontal lines. You can see both those chips (covered in a glob) one behind the screen in this image, and the other on the flexible PCB below the screen:

Sources:

• The block diagrams are from the Game Boy Programming Manual and patent 5134391
• Thanks to NekoNova for posting a really good picture of the PCB on modretro thread 11852 and this Russian post for the other one
• Thanks to richy486 for the excellent photo from his backlight tutorial
• Thanks to marc rawer's project page for the pinouts
• A game boy color PCB from wikipedia 
• http://gbdev.gg8.se/wiki/articles/DMG_Schematics