I've worked with a number of different mediums for oil painting over the years. There was a time that I was very fond of Stand Oil which is a thicker version of Linseed Oil. It is almost like honey it is so thick! There are mediums meant to increase the fat level of the paint and mediums to increase transparency. There are mediums to decrease the drying time. The most useful medium for oil painting that I have ever found is Liquin by Winsor Newton.
Liquin was original developed for WN's alkyd line of paints. Alkyds work very similarly to oil paints but they dry more quickly. I used to only paint with alkyds. Winsor Newton tried to popularize the alkyd paints, but they never really caught on so WN discontinued the artist grade version. That is when I stopped painting with alkyds.
Liquin is completely compatible with oil paints. It speeds drying time, increases flexibility and it can also be used for glazes. Liquin is not an oil as is Linseed Oil so it doesn't actually add "fat" to the oil paints, but it does increase flexibility which is the same as increasing fat! It's wonderful stuff!!
If you've never tried it, do! Just buy the small jars! Unless you are using huge quantities of Liquin all the time, small jars will last for a long time and won't get "wonky" by being in a jar that has increasing quantities of air every time you use it.
Great stuff - don't paint without it!!