When repairs on appliances are necessary, they can be costly. Here is a list of appliance types and my opinion of which ones you should consider purchasing an extended warranty for.
Built-in Models
When a major problem arises, it may not be a simple matter of just going out and getting a replacement. Most built in appliances (except dishwashers) are not standardized in size and cabinet work may also be required for a replacement.
Electronic Models 
Appliances with electronic controls can be difficult to diagnose and expensive to repair. If you live in a rural area with frequent power fluctuations or outages that can play havoc with electronic controls, you should probably look for a simpler design altogether.
Be aware that many of the energy efficient and high-end washing machines and dryers have electronic components that may not be apparent at just a glance.
Machines don't actually need to have a touchpad or LED screen to contain electronic components. Click on the image above for an example.
RefrigeratorsRefrigerators are the most expensive appliance to repair. In addition, some jobs are not possible for home-owners to complete themselves because special tools and training are required.
Gas RangesGas ranges are next. Models with electronic ignition and/or electronic controls have parts that are expensive and sometimes difficult to diagnose. Also, models which use a 'hot surface' (glow bar) ignition system for the oven (the most common type) will likely need to have the oven igniters replaced several times over the course of the product's life span.
Standing pilot 
ignition models, on the other hand, are fairly simple and generally require fewer repairs. Additionally, all burners (surface and oven) will still be functional in the event of a power failure.
Electric RangesIf you purchase a self-cleaning or electronic range, I suggest that an extended warranty policy be on your shopping list too. These are complicated appliances with parts that tend to be expensive.
However, if you are purchasing a basic electric range with regular coil surface 
elements, most repairs are simple enough that handy home-owners can do themselves.
Automatic washersMany automatic washer problems can be difficult to diagnose and if the repairs get put off for any length of time, they can become expensive and difficult to fix. An extended warranty will allow you to get it fixed without delaying until you have the funds available to pay for the repairs.
Most front loading models are considerably more complicated than top loading models and are usually more difficult and costly to repair. Additionally, all the models I know of have electronic components that can make problem diagnosis difficult for a novice or homeowner to repair themselves.
Electric DryersMost low to mid-line electric dryers, on the other hand, are relatively easy to repair. They have simple mechanisms and have very little wiring. Parts are usually not expensive and are available at most parts outlets.
Top of the line models may have electronic controls and related components in them in which case an extended warranty may be a good choice.
Gas DryersGas dryers should be serviced by qualified technicians only. Heating problems can be difficult to diagnose and dangerous if not corrected properly.
DishwasherDishwasher repairs should also be a job for a professional. When you mix water, electric and mechanical components, all sorts of problems can occur with improper service. In addition, major parts are expensive and can be difficult to diagnose.
Freezers
Air Conditioners
