All known hadrons (composite particles made of quarks) have two quarks (mesons) or three quarks (baryons). Mesons are bosons which come in pseudoscalar (spin 0), or vector (spin 1) varieties. Baryons come in spin 1/2 and spin 3/2 varieties. The only stable hadrons are protons and bound neutrons, both of which have spin 1/2.
There is some evidence of pairs of mesons bound into a "molecule" although it isn't entirely clear if the concept of intrinsic spin applies to this system. Unitary tetraquarks and pentaquarks (as opposed to "molecules" of two mesons, or of baryons and mesons) have been theorized but not observed.
There is also a theoretical variant on a hydrogen atom in which a antimatter version of a charged lepton takes the place of a proton, such as muonium. A number of other "exotic atoms" have been conceived.
This leaves some hypothetical loose ends.
I've discussed before the possibilty of a particle made of two neutrinos of different flavors and opposite matter/antimatter status bound by the Z boson mediated weak force into a "molecule". If this functioned as a composite particle, these would be pseudoscalar spin 0 particles (and like kind neutrinos would repel each other rather than bind, preventing the formation of the analog to spin 1 mesons), but it isn't clear that intrinsic spin would apply to this kind of bound system.
Another interesting possibility that is well motivated theoretically is the possibility of glueballs. These strong force bound particles should be capable of existing as free particles, like other hadrons, and should have well defined intrinsic spin, which could be either spin-0 (scalar or pseudoscalar), or spin-2 (tensor) (presumably each made of two gluons). Presumably, there could be either two gluon or three gluon states that would be color charge neutral. Conceptually, spin 1 (vector) and spin 3 glueballs (made of three gluons) are possible.
In particle physics, a glueball is a hypothetical composite particle. It consists solely of gluon particles, without valence quarks. Such a state is possible because gluons carry color charge and experience the strong interaction. Glueballs are extremely difficult to identify in particle accelerators, because they mix with ordinary meson states.
There have been hints of direct glueball detection as far back as 2005 (see also here), but even now, there is not a definitive detection.
As a 2008 review explained:
Many different experiments exploiting a large variety of production mechanisms have presented results in recent years on light mesons with J(PC) = 0(++), 0(-+), and 2(++) quantum numbers. This review looks at the experimental status of glueballs. Good evidence exists for a scalar glueball which is mixed with nearby mesons, but a full understanding is still missing. Evidence for tensor and pseudoscalar glueballs are weak at best.
Since some glueballs appear in lattice simulations to be massive (on the order of 1-2 GeV/c^2) and scalar, a scalar glueball would look very much like the hypothetical Higgs boson, but lighter.
As recently as 2011, published physics papers have contemplated what sort of experimental signature a pseudoscalar glueball would have and compared it to the high energy physics evidence.
While some particles of spin 0 (scalar and pseudoscalar mesons and perhaps glueballs and hypothetically Higgs bosons), 1/2 (fermions and some baryons), 1 (fundamental bosons and some mesons and hypothetically some three gluon glueballs) and 3/2 (some baryons) have been observed; no particles of spin 2 (hypothetically gravitons or two gluon tensor glueballs, some candidates for which has been identified, and also here) or 3 (hypothetically some three gluon glueballs) have been observed.
There are questions, however, about whether gluons are even a meaningful way to describe the low energy limits of QCD. For example, there are doubts about whether intrinsic gluon spin or gluon helicity is the proper concept to use to theoretically model glueballs. Theoretical challenges to even the existence of three gluon glueballs are particularly daunting, although some progress has been made in modeling them in lattice QCD models.
Then, there are gluelumps (also here) which are "one-gluon states, allowed because of the static colour octet source.", in more elaborate models such as SUSY ("These are another hypothetical type of gluonic hadrons, where a spinless, static, color-octet source is added to the gauge field. Gluelumps have actually been a first attempt to model gluino-gluon bound states.") Another of the references talks about a gluelump made of a charm-anti-charm and gluon triplet.
Given the fact that all of the fundamental bosons and at least one kind of meson act as force carriers phenomenologically, and the fact that all glueballs are bosons composed themselves entirely of bosons, the possibility that glueballs might act as force carriers of some type is fascinating. We know that it is conceptually possible for a spin-2 particle to reproduce gravity (and indeed that one can understand QCD through a correspondence between its gauge theory and five dimensional classical gravity in anti-deSitter space), but what kind of force might be carried by a spin-3 glueball if one existed?
Some consideration has been given to this question, and spin-3 bosons present fewer theoretical challenges than high spin bosons (see also here). This has some application in condensed matter physics and might matter in SUSY theories.