Possibility of "graviballs"? Looking at the relevant wikipedia page, one can read that the graviton should be massless. Is it 100 % certain that it is massless or is there room in any "nonstandard" models for a tiny non-zero mass (which could lead to a similar surprise as the detection of the neutrino oscillation) such that the graviton (if it exists ...) could maybe selfinteract and form something like "graviballs"?
(My present knowledge of GR and QFT is at the "Demystified-Level")
 A: Gravitons do self-interact. That's because gravity couples to energy-momentum tensor which is non-zero even for massless particles (e.g. consider that gravity influences light). So it's certainly a possibility that they might form a bound state. But you can't see this in a linearized theory of gravity because it is a free theory (similar to electrodynamics). So you'd also have to include higher order curvature terms and quantize those, which would in principle resemble something like chromodynamics (although much harder). It is also not clear whether the theory would be consistent without including the rest of the standard model (and beyond). In any case you'd definitely need some form of quantum gravity to answer this question.
Regarding the first question: there are experimental upper bounds on masses of all massless particles. Obviously it's impossible to distinguish whether something is strictly zero or infinitesimally small. Suffice it to say that all experiments conform to the fact that the masses are zero and there is no theoretical justification whatsoever to consider other models.
A: There exists a self-consistent, self-interacting gravitational solitons called "gravitational geon" within general relativity. As Marek said above, gravitational field is non-linear and self-interacting so energy momentum of gravity wave itself will produce bound states. You can read a bit more about it in a following Wiki page:
http://en.wikipedia.org/wiki/Geon_%28physics%29
One important contribution that the Wiki page missed is that you can get a half-integer spin geon from pure gravity, a work by Friedman and Sorkin.
http://prl.aps.org/abstract/PRL/v44/i17/p1100_1
As far as I know there are not much development in quantum mechanical aspect of them.
A: Two recent papers in that direction:
https://www.sciencedirect.com/science/article/pii/S0370269322001253?via%3Dihub
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.081601
