One of the most notorious unjuicables is the banana.  Bananas are too soft, and contain little juice compared to other fruits.  It is very difficult for a juicer to separate the little juice in the pulp.  This will cause your juicer to clog, and will leave a bunch of mush that is a pain to clean.  Bananas go great with many fruit juicer recipes, so it is unfortunate that you cannot throw it in your juicer.  However, if you have a blender handy, you can juice all the other ingredients separately, then blend the bananas with the juice.  You may or may not find it to be too much extra trouble though.

Avocados are another troublesome fruit.  They have a buttery texture that is very soft and contains very little juice, which will create a pasty mess to clean up.  Again, you can get around this problem with a blender, but there are very few juicer recipes that need an avocado anyway.

Leafy greens can be juiced, but depending on your juicer, it may not be worth it.  They contain very little juice, so it would take an auger juicer or a gear juicer to get the most out foods like spinach or broccoli.  Centrifugal juicers will have a tough time with leafy greens, though it will suffice for getting some greens in your juicer recipes.

Other produce include coconuts and mangos.  I have not ever tried juicing a coconut, but some sources say that you cannot juice a coconut.  Generally, mangos can be juiced, but it depends on how capable your juicer is.  Personally, I have never had a problem with mangos with my Omega 8005 auger juicer.