I would probably have to say running the plumbing. There are alot of strict ways to install plumbing on a addition with permits and inspection. Make sure you have a very good plumber contractor to complete the job right the first time.
Matching the exterior is only difficult if you have a brick or stucco exterior. Other materials are much easier.
Plumbing is difficult if you do not have existing plumbing near the addition and only the sewer lines are truly difficult. Water is much more flexible to run.
Electrical is not usually very difficult unless you need a larger electrical service panel.
HVAC can be difficult, depending on your existing system. Hot water is more difficult than forced air, but electrical is fairly simple.
Electrical and HVAC have to conform with Building Codes. Though stucco exterior also has to follow building standards, the texture and finish, including the color match, requires specialized expertise.
Running the plumbing is one the most complicated aspects of an addition. The electrical can be easy if you have space on the panel for the new lines of the addition. If you don’t have space in the panel, it may by expensive because you may have to upgrade the panel as well. The HVAC can be expensive if your existing unit is not big enough to cover the new addition. Matching the exterior may vary, depending on what type of exterior you have.
Taking the time to do it right, pull a permit, pass inspection, conduct proper heat load calculation and design/size/install properly and deliver a good service to client and properly engineered system.
Multi and variable stage compressors require additional time, experience, maintenance and training.
Although there might be few technical challenges to build an addition, the most important one will be to make it look like it is not an addition and it was designed this way, which is an accumulation of many finish details that the contractor should pay attention to.
As far as the more technical aspects of an addition, running new plumbing can be the most complicated aspect especially if there are no existing sewer or drainage lines where needed. With regards to aesthetics, matching the exterior can get pretty difficult given the the end goal is for the house to look as cohesive as possible and not like there was something added to the house. At that point, the contractor has to make sure they are paying close attention to detail to achieve that.
I would also say that the plumbing could present a real challenge for the sub contractor when the sewer line run is stretched longer than usual