Lots of excellent advice to be found in the comments, I won't stress anything aside from agreeing that these small engines shouldn't take more than 3 pulls to start. After 5 or 10 pulls you'll save yourself frustration by just stopping there and taking the plug out to see if you've flooded it or if it's not getting fuel (or spark). If it started easily in the past, and now find the plug black, oily and wet looking, buy a new plug.
If the plug is dry and the primer bulb is functioning okay, the carb probably has become gummed-up. A little trick that sometimes works for me (I work on lots of little 2-strokes that use these carbs) is to remove the air cleaner and shoot a 1/2 teaspoon of fuel into the carb's intake using an eye dropper. It's enough to keep the engine running momentarily and MIGHT suck the gummy crap free. If so, and there was gas left in the tank, dump it and mix up an all new fresh batch. Before resigning myself to rebuilding a carb I'll first dump all the gas and put just enough "Seafoam" in the tank to get it through the primer system. I'll let that sit while hoping it'll work it's magic... I'll then go back and attempt to start it using the force-feeding, eye dropper technique ~but add 'some' mixed fuel into the tank FIRST or else you might be so happy it's running that you forget the importance of the fuel-oil mixture!