I have assembled so many book shelves, office desks, chairs, tables, entertainment centers, computer desks...basically anything that comes in a box I have assembled, and probably more than once. But I wouldn't really consider that being a handyman.
I am a quite capable handyman, with building two outside decks, installing and replacing many windows and doors, installed siding, changed roofing shingles, built kitchen cabinets from scratch, installed laminated, hardwood and ceramic flooring, etc...so many projects that I could go on.
I have dabbled with electrical and plumbing but I am not as confident with those two. I am capable of remodeling a bathroom right down to the floor joists (which I have done) but I wouldn't setup and install plumbing to a whole house. Same goes with electrical work. I can replace a fixture or an outlet but I wouldn't rewire my house.
I have just about every hand and power tool a person needs to get most projects done. Of course I don't have everything but that's when I decide if I need to add it to my collection.