The Skyrim Oghma Infinium glitch is an exploit that can be used to easily increase skills in Skyrim. As such it is a potentially game busting/ruining bug. To take advantage of the Oghma Infinium glitch, one needs to have completed the quest Discerning the Transmundane and obtained the Oghma Infinium book that is the daedric artifact related to Hermaeus Mora. One also needs to use a container for the trick which may differ depending on the version of Skyrim one is playing. In earlier and later versions a bookshelf may be required while version 1.2 of the game may use any containers like chests, cupboards, and dead bodies but not a bookshelf. I can confirm that in an earlier version that the bookshelf that comes as a decoration improvement for Breezehome in Whiterun can be used for this leveling trick. The version 1.9 patch supposedly removes this Oghma Infinium bug. It is highly recommended to save before experimenting with this exploit.
Using the Breezehome bookshelf this is how the bug works:
1. With the Oghma Infinium in your inventory activate the bookshelf
2. Using the bookshelf menu go into your inventory to look at the unread Oghma Infinium. Options should appear.
3. Select an improvement path (Path of Might, Path of Shadows, or Path of Magic)
4. Go back to the bookshelf menu by pressing the back button once (usually Tab key on a PC)
5. Place the Oghma Infinium on the bookshelf by selecting Store (usually R key on a PC)
6. Exit the menu. The Oghma Infinium should appear on the bookshelf.
7. Select the Oghma Infinium (not the bookshelf). The Oghma Infinium should automatically open up and present an option screen.
8. Select Do not read.
9. Select Take (usually E key on a PC).
10. Go back to step 1 and repeat as often as desired. One can max out all skills at 100.
Make sure to perform the above sequence strictly and slowly if the option menu is not popping up after the first time or the Oghma Infinium is simply disappearing. Difficulty obtaining the desired effect was noticed when going too fast. Difficulty was also encountered when confusing/conflating activating the bookshelf with activating the Oghma Infinium, one may lead to a different result from the other. Also notice there is no intermediate step between step 8 and step 9 such as closing the book; the book is taken directly after choosing Do not read.
If reproducing this exploit is proving difficult despite following these directions on how to perform the Oghma Infinium glitch, try loading another save or exiting Whiterun and returning. One might also try placing the Oghma Infinium on the bookshelf first then start the cycle from step 7.