I do not work on many Chevys but most new cars have eliminated grease fittings and sealed the joints for life. Most aftermarket replacement parts are also sealed for life but I look for MOOG replacement parts because they usually have grease fittings.
Grease fittings may be present on the ball joints upper and lower on both sides of the front suspension, on the front and rear u-joints on the drive shaft. Sorry I can't be more specific hopefully some one else will reply or if you find out how many or where they are if you could post our findings it may help someone else in the future.
I cannot say for sure without seeing the underside but my best guess would be that you have 10 grease fittings under your 2006 Suburban - I cannot say for sure because sometimes they don't include grease fittings in one component or another. Here is where I would look for fittings. Upper and lower ball joints (4), Inner and outer tie rod ends (4), Pitman arm (1), and idler arm (1).