Is there a problem with our currently known fundamental particles? Many scientists are searching for particles even more fundamental than leptons, quarks, gluons, etc. and (from what I know) string theory tries to hypothesize one elementary "thing" that everything is made of. So is there a reason that we should think that our currently known elementary particles may not be the true elementary particles of the universe?
 A: Some scientists do believe there are more fundamental particles (or strings, in the case of string theory). Others believe it isn't so much that there are more fundamental particles, it's that there are fundamental particles not yet discovered (as an obviously theoretical example, the graviton). 
As for the reason, the current theory of physics (the Standard Model) doesn't explain nearly everything and has some problems! Examples (to name a few):


*

*it doesn't explain gravity

*doesn't address dark matter

*doesn't explain dark energy

*physicists call it inelegant: there are 19 "arbitrary" constants

*the hiearchy problem

*doesn't fit with a lot of discoveries in cosmology


I could go on (there are more here).
Finally, as CuriousOne said in the comments, every time we've sharpened our view on the world, something new and unexpected has popped up, from when Galileo first turned the telescope to the stars, providing proof of Copernicus' theory, to when Rutherford found evidence for antimatter in cloud chambers, to when two teams of physicists at CERN found evidence for a new particle that turned out to be the Higgs boson. Humanity is forever finding something new, and especially now, when we are finally exploring the outer reaches of the universe with LIGO and space telescopes, and probing the subatomic world with CERN, Fermilab, and others...there's no reason to think that will change. 
Hope this helps! 
