The recycling logo "“ those three arrows going around in an endless loop "“ is lying to you. While glass and plastic can be recycled indefinitely (at least in theory for plastic, which is usually "upcycled" or "downcycled" into products that can't be recycled), paper only gets so many go-rounds through the circle of life.
On trash day, the paper in our recycling bins goes to a recycling center and then a paper mill, where it's put into vat called a pulper. The pulper, which also contains water and some chemicals, is essentially a giant blender and shreds the recovered paper into small pieces. The paper-liquid mixture is then heated so the paper breaks down into little fibers. The resulting mush is called pulp.
Contaminants are removed from the pulp by cleaning and screening. The pulp is cleaned in a spinning cone-shaped cylinder that separates heavy contaminants like staples. Screening is exactly what it sounds like. The pulp is forced through various screens to remove smaller contaminants like bits of glue.
Some recovered paper is de-inked, and some is not. De-inking can be done in two ways: washing or flotation. During washing, chemicals are added to the pulp to separate the ink from the paper fibers. The pulp is then rinsed with water and the ink is washed away. In the flotation method, the pulp is put in another water-filled vat called a flotation cell. Surfactants are used to loosen the ink from the pulp, and air is passed through the cell. The air bubbles float the ink to the top of the mixture and the resulting froth skimmed off the top.
The pulp is then beaten to separate large clumps of fibers and make the fibers swell so they're better suited for papermaking. At this point, if white recycled paper is being made, the pulp is also bleached to make it bright white (if brown recycled paper is being made, say, for industrial paper towels, the pulp doesn't get bleached).
To make the pulp back into paper, the recycled fibers are blended with new wood fiber (virgin fiber) to give it extra strength and mixed with water. A paper machine sprays the watery pulp in a wide jet onto a wire screen conveyor belt. As the pulp travels on the belt, water drains through the screen and the paper fibers begin to bond together. A series of press rollers squeeze out more water and heated metal rollers dry the paper, which, now finished, is wound into a roll that can be cut into smaller rolls or sheets.
Phew. As you can see, paper goes through quite a bit of abuse during the recycling process and each time recovered paper goes through the whole song and dance the individual wood fibers become shorter and more brittle and lose their strength. Paper industry folks say that the average paper fiber can survive recycling six to eight times before its short enough to slip away during the screening process.