Update 2017: The stock youtube app now gives you a few options for playback rate when you click on the three dot manu when playing a video, for most phones.
One Way
Download an external player that lets you control the speed, like BSplayer.
Then in the youtube app when you get to the video you want to watch click the Share button and then select the player you want to use. The external player will open and it will play the video you want, and you can control the speed in it.
Tip: you can also use this method to skip ads, if you share the video to a different player you can watch it right away without having to wait if there's an ad.
Edit: I mentioned VLC and Dice Player as other options to control video speed, but looks like I can't share a video from the youtube app to play with them, at least on my phone
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Alternative Way (not as smooth):
This works for watching videos on m.youtube.com on the Mobile Chrome browser (probably other browsers as well)
Create a bookmarklet, name it something like 'yspeed', and use this code:
javascript: document.getElementsByTagName("video")[0].playbackRate = parseFloat( prompt("Enter Rate") );
then in your phone when you're watching a video on m.youtube.com, in the url bar type 'yspeed', don't press enter, you should see your bookmarklet popping up as an option. When you tap it you will see a popup to enter your speed (.5, 2.5, 3, etc)
Speeding videos up this way might get laggy, and the max speed you are able to watch it smoothly might depend on your device capabilities
How to create a bookmarklet:
On Desktop
- Right click the bookmarks bar
- Click Add Page
- Edit the name, and type the javascript code (including "javascript:") into the URL field
- Select your Mobile Bookmarks folder (might be optional) and then click save
On Mobile Chrome
- On any page (that you haven't bookmarked already) click the three dot menu button next to the address bar
- Tap the star
- Tap Edit on the popup that shows up on the bottom of the screen. If you missed it then tap on the star again and it will take you to the Edit menu.
- Enter the name, select your Mobile Bookmarks folder, and replace the content of the URL field with the javascript code (including "javascript:")
Some more info on bookmarklets: http://www.labnol.org/software/google-chrome-bookmarklets/27894/
Pro Tip: You might get a less laggy video if you watch the video in your phone browser in Desktop Mode and then select a lower resolution video. You can also select speed there through the youtube controls, but if you want more options you can use the bookmarklet.