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Definition of incompatible : 1. An incompatible substance; esp., in pl., things which can not be placed or used together because of a change of chemical composition or of opposing medicinal qualities; as, the incompatibles of iron.
2. Incapable of being together without mutual reaction or decomposition, as certain medicines.
3. Not compatible; so differing as to be incapable of harmonious combination or coexistence; inconsistent in thought or being; irreconcilably disagreeing; as, persons of incompatible tempers; incompatible colors, desires, ambition.

Synonyms:

antonymous, mismatched, absurd, antipathetical, hostile, conflicting, clever, automatically, unadapted, unfriendly, irreconcilable, contrary, agree, at variance, broken, jarring, clean, incoherent, immiscible, built, distinct, contrastive, disagreeable, unsuited, contradictory, mismated, auxiliary, analog, ill-sorted, inharmonious, dissonant, inadmissible, contrasting, unmixable, diverse, disagreeing, discordant, discrepant, unfitting, mutually exclusive, clockwork, inconsonant, opposed, antipathetic, dissimilar, ill-matched, alternative, unlike, repugnant, antagonistic, unsympathetic, out or keeping, various, uncongenial, cordless, inapposite, factious, incommensurable, incongruent, automatic, opposite, unsuitable, clashing, counter, inappropriate

fey, quirky, curious, funny, uncustomary, bizarre, unusual, different, incongruous, inconsistent, unconventional, freakish, aberrant, un-ordinary, odd, divergent, off, crosspatch, crazy, misfit, deviant, outlandish, atypical, oddball, singular, novel, extraordinary, uncommon, deranged, oddity, queer, improper, anomalous, idiosyncratic, mismatch, strange, eccentric, nonconforming, out of the ordinary, untypical, erratic, unique, exceptional, abnormal, peculiar, irregular, unfamiliar

Usage examples:

• In 1813, the Cortes- general of the kingdom adopted the same measures, after declaring that the existence of the privileged tribunal of the holy office was incompatible with the political constitution which had been decreed, published, and received by the nation. - "The History of the Inquisition of Spain from the Time of its Establishment to the Reign of Ferdinand VII.", Juan Antonio Llorente.
• The combination of these two results must bring down the temperature of Mars to a degree wholly incompatible with the existence of animal life. - "Is Mars Habitable?", Alfred Russel Wallace.
• While our supreme interests demand the exclusion of a laboring element which experience has shown to be incompatible with our social life, all steps to compass this imperative need should be accompanied with a recognition of the claim of those strangers now lawfully among us to humane and just treatment. - "Complete State of the Union Addresses from 1790 to the Present", Various.