Well - this depends on how broadly you're defining "organism" ;)
Some things that might be possible:
Non-gametic sexual reproduction - I don't know a good term for this, but basically, development of an individual with the genes of two or more parents through a mechanism other than the normal one (fusion of gametes into a zygote which then develops by cell division).
For example, both parents might contribute a large number of cells which then assemble (either 'autonomously' or through help from the parents) into a smaller individual. This would work best in a "proto-multicellular" lifeform where the cells retain some degree of independent survival ability. In such a species, every individual would be a chimera (different genes in different cells).
'Scaffolding' or 'crystallization' - the parent organism physically forms a "scaffold" for the 'assembly' of the new individual, or the simple presence of a parent organism induces a "substrate" material to form new individuals. Distinct from budding / fission because the new individual's biomass doesn't derive from the parent.
This is roughly how prions work in the real world (a misfolded, pathogenic prion protein induces normal prion protein to rearrange into the pathogenic form) though admittedly they don't have a metabolism and thus aren't technically alive.
Some similar things have been suggested with regard to "self-assembling" and "autocatalytic" forms in speculative hypotheses on the origin of life - PAH world (polycyclic aromatic hydrocarbons as a precursor stage to RNA in the origin of life) and iron-sulfur world hypothesis.