While I would love to see one, I have never seen a ball of yarn large enough to create an entire sweater or a skein with perfectly placed color changes, so you never have to join a new ball of yarn. When I first learned to crochet, I didn’t know how to properly join a new ball or color. I would fasten off the first ball, create a slip knot, and join the new ball. Or I would knot the two ends together. Either of these methods created a knot in the fabric and can be an annoying, though small, waste of time. When you have spent hours on a handmade creation, little imperfections such as these hard knots can be disheartening.
There is a different option. You can join a new ball or color as you go without fastening of the first color. It's surprisingly easy actually. I even snapped a few pictures to illustrate.
Work the stitch before the color or ball change to the last two loops. I used single crochet but the same instructions apply for double crochet, treble crochet, a puff stitch, or about any other stitch you can think of.
Now, instead or yarning over and drawing the old color through, yarn over with the new color and draw it through the two loops. It can be a little tricky coordinating all of the tails at first, but it become second nature. Until you work a few stitches the new color will be a bit loose; be careful not to pull on it as that will shorten the tail which will need woven in later for stability.
And that's it, you have started a new color or ball without the annoyance of a knot or fastening off and rejoining. I would love to hear or discuss other methods of joining new yarns or switching colors while crocheting.