Words that have identical vowel-based rhyme sounds in the tonic syllable. Moreover, that tonic syllable must start with a different consonantal sound.
There are no pure rhymes for "rachel o'brien".
Words that have a pure rhyme on their last syllable only.
purchasin'
    repurchasin'
      rationin'
        stationin'
          vacationin'
            regulatin'
              deregulatin'

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                Words that "almost" rhyme on the vowel-based rhyme sound of the stressed syllable like: be/eat or maybe/shapely.
                rationin'
                  stationin'
                    vacationin'
                      lengthenin'
                        strengthenin'
                          stabilizin'
                            destabilizin'

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                              Rhymes made up of more than one word. For instance, "jealous" and "tell us" or "shaky" and "make me."
                              There are no mosaic rhymes for "rachel o'brien".