In Silas Marner, what is Godfrey’s great unfulfilled desire?
I assume you are referring to Godfrey in Book II of this incredible novel rather than Book I. In Book II, of course, Godfrey has received his "lucky escape" of being able to rid himself of his unfortunate choice of first wife without people discovering the truth about him. He has since married Nancy, whom he has desired for so long, and when Book II opens, sixteen years have passed between the end of Book I. However, as Chapter Seventeen makes clear, the one element that prevents perfect felicity in the marriage between Nancy and Godfrey is that Nancy is unable to bear children. This has led Godfrey to suggest to Nancy that they adopt, and in particular adopt Eppie, his legal daughter, but Nancy is reluctant to agree to this proposal, and we see Nancy questioning whether she was right to reject this idea:
Had she done everything in her power to lighten Godfrey's privation? Ha she really been right in the resistance which had cost her so much pain six years ago, and again for years ago--the resistance to her husband's wish that they should adopt a child?
Note how this quote identifies the "privation" that Godfrey is suffering at having no heir, and also the probable guilt that he feels when he sees his rightful daughter that he has disowned to all intents and purposes. "Adopting" Eppie would allow him to appease his guilt whilst also giving him a child to call his own and not revealing his shameful secret.