A stronger word than incompatible, which suggests that two things can't be with each other ("not congruent" as one dictionary puts it) -- incompossible suggests not just they can't be in proximity but can't both exist at the same time. Not rare if not obsolete, which is a pity as sometimes you want a stronger word than incompatible. From Latin incompossibilis, but possibly under the influence of French incompossible -- the roots break down as in-, not + com-, together + possibilis, possible, from posse, to be able.
I told my kids last week about the engagement -- Brad just sorta shrugged but Charlene still refuses to accept it, insisting that she and Felicity are incompossible and she'll never accept her as her mother.
Administrivia: I won't be able to post next Thursday -- can any of our eager volunteer backups take over for the 7th of October?