Contrary to what is commonly believed, perhaps we can build a space elevator with existing materials. The point is that we don't think about how to build this structure in a really appropriate way, as the authors of this paper or this other paper claims.
The structure should have elasticity to withstand the tidal forces of the Moon, which will have some kind of replacement due to the attack of micrometeoroids, shields to protect from radiation, protection against the actions of the atmosphere, support of solar panels throughout the entire structure , piping for water, oxygen, nitrogen , basic structure material, plus exclusive service rails for AI-controlled machines that will take care of repairs, different tracks for getting on and off, which do not really need to be different for cargo and passengers, in addition to perhaps a possible expansion of the structure in order for new tracks to be built.
Anyway, this structure will be huge in itself, even without stopping stations.
You aren't going to make a space elevator economically interesting to build as long as the volume of cargo can be taken into space or brought to Earth through other ways , so if you have reached the point of building one or more space elevators, it is because your society it has a big trade between the sides of the atmosphere.
All of these loads going up and down through the elevator will generate forces that the cable must resist and balance, which requires that it be dimensioned much higher than what would be simply a string of diamond nanotubes, graphene, carbyne, duct tape or any other supermaterial.
Here is an interesting point: if the load moves too fast, you will not need many intermediate stations, but the forces of this whole movement will require an oversize of the cables. If the load moves slowly you will need more intermediate stations, and they will require more of the cables as well.
In conclusion, any type of intermediate station, with some type of hotel with panoramic views, maintenance of the cars, emergency services, escape pods, restaurants, etc., will not present a significant effort in the structure of an elevator, unless you want to do a small town.
 I don't really know why most science fiction stories or Mars terraforming suggestions ignore the importance of nitrogen in atmospheric composition.
 methods that are already and will be in use by society then, the proposal for a space elevator will always looks like radical until its construction be an inevitable necessity.
 user535733 deserve all upvotes for mention the last book of Clarke's Space Odyssey.
 I failed in learn how make decent footnotes here, someone please edit it