Right from the beginning of the book, the children, especially Dill, are fascinated and petrified of Boo. I am not sure that how badly they want to get him out. We need to keep in mind that they were scared of him.
What the children do is go by Boo's house out of curiosity. On one occasion, they get in a tire and Scout rolls into Boo's yard. On another occasion, they trespass and Jem's pants get stuck in the fence. And still another time, Jem runs up to touch the house and runs back.
Are these ways the children want to make Boo come out? It is debatable.
As to how many times Boo came out, this cannot be answered with any degree of certainty. All we know is that Boo came out. For example, he is the one who put the gifts in the tree knot. He also got Jem's pants and fixed them. He also put a blanket around Scout's shoulder while Ms. Maudie's house was burning. And finally, he was watching them when Mr. Ewell attacked.
In light of this, we can say that he came out many times.
I agree with the above answer it is debatable on the number of times Boo came out of his house. I think the most notable time and the one easiest to pick out is when Boo comes out of his house when he helps Scout and Jem at the end of the novel.