It's the way it's usually done. A 6 string guitar is close to 1" after putting it together, but then the backside is carved and the fretboard gets a radius. The first fret area will usually be rounded and the higher end will usually be flater to enhance hand comfort/control and ease of play.
A 4 string or 3 string has a neck that's not as wide. That gives easy finger access all the way over the fretboard. Some prefer a large mass for the entire neck for comfort/control on these necks. It just comes down to what is more comfortable for you and gives you the control you need.
I have medium length and width fingers and find that the 3/4" necks with 1/4" work well for me. I usually carve a V shape to the back of the necks and use a radius on the fretboard because I'm used to that from playing 6string guitars for many years. Some prefer a squarish neck with rounded edges.
One inch is "thick" for a neck at the nut end, but not down at the box end. Without a truss rod, the additional beef at the nut end may be warranted (or not depending on the string selection). Most players/builders keep it simple and just play with the neck at a constant thickness (1"). Since the neck is only 1 3/4" wide, it's not like trying to reach around a log.
There are several things you can do regarding the neck profile, depending on the wood being used and how much work you want to put into it.
It is possible to taper the neck from nut to box and round the neck contour.
Another thing that you can do is use a fret board that is a bit under 1/4" thick. There is a practical limit to how thin you can make it based on the frets you use, but once it is glued to the neck, it will be strong and rigid.
A final thing is to taper the neck from the nut to the box in the up/down direction. This is pretty much what a traditional acoustic guitar has. My Gretsch acoustic is only 9/16" thick at the nut (including the fret board).
While there are many things you "could" do. I would suggest that you begin with a normal 1 3/4" wide neck and a 1/4" fret board. Fully round the underside of the neck. Make it nice and smooth the play it for a while and see if an ultra-thin neck is really worth the additional work.
Keep us posted
Tom's reply shows how people on this page are willing to share their knowledge.
If it is a 3 string and both woods are hardwood, 3/4" is plenty thick in total. Built and seen plenty of 3 stringers in oak or maple with no fretboard and no issues. Poplar and other soft wood is another story. Whether it is worth the effort to thin it down is the next question! You can always build it, then round it or thin it more if you don't like it, but it is not so easy to add some back.