Many of such words would be quite technical, not "primitives" or least-common denominator; and would not be essential to a " basic" vocabulary. obviate, parallel, isosceles, gastroenterologist, pahoehoe, anemometer, sphygnomometer, pyroclastic, enantiomer.... These are not "irreducible", but it takes more than one word to " reduce" them. (of couse, now someone will come up with one-word synonyms just to refute my offhand examples).
I would suspect that invented languages would have a lot fewer synonyms than English, which has in many cases inherited or adapted multiple terms for the same thing from various languages.
Here's an article about Ithkuil, an invented language ostensibly much more efficient (more precise, yet more concise) than English. http://www.newyorker.com/magazine/2012/12/24/utopian-for-beginners
I've been looking for a list of words that don't have synonyms. I've had no luck. I have however researched the idea for about two years now and the closest I have gotten to previous work on the matter was by the linguist Morris Swadesh. He is a little dated on his work. Wikipedia has an appendix of "Swadesh lists" in various languages: http://en.wiktionary.org/wiki/Appendix:Swadesh_lists
These words tend to lack synonyms in their respective languages. Words like colors or body parts. There are verbs as well though.
I have recently become interested in words, mainly due to a fandom I am in where words play a big deal in that world and stuff, and I have been trying to gather some unique words so I could expand my vocabulary without it being another "hey, you know that word everyone uses, here's a word that means the same but it unpronounceable". I've managed to gather some unique words, such as Apricity which means "the warmth of the sun during winter" which is interesting due to the severe contrast, and also Sonder which refers to "the realization that everyone around you, even mere strangers, have lives as full and real as your own where they are the central character" which is quite a specific word.
I am interested to expand my vocabulary without it just being a useless word due to it being another "hey, here's a word for a hooligan, or here, here's another word for stupid", when like no one is going to use that, they will just use the everyday word.
Wait, is that the right tag to give this? Like, you read words, but is that enough to put into this tag? Apologies if it is not the right one.