Words without simple synonyms
Solution 1:
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
Solution 2:
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.