Whisk eggs without an electric whisker
Yes, you can whip egg whites (or whipped cream, or....) by hand. There are a few things you need:


a reasonably large bowl
a good, sturdy whisk, again not too small
proper technique
a good amount of ellbow grease
patience


It will typically take longer than when using a mixer (for beginners, I've seen pros that could keep up with any measly old mixer, especially taking cleanup time into account) but lets you control the outcome better: less chance of overmixing or uneven results.

I myself have done it several times, either because I had no mixer at hand or because I was working in the middle of the night and didn't want to wake everyone and the neighbours.

For motivation, see this video and this video for technique and this video for fun or any other tutorial on the web.