What is meant by ‘threefold bliss’ in the “verses of sharing and aspiration”
This page ...


  2007-11-16 The Threefold Bliss 55:28  
  
  The talk explores worldly or human happiness, celestial or heavenly happiness and transcendent happiness or Nibbana


... says that it means:


  
  worldly or human happiness
  celestial or heavenly happiness
  transcendent happiness or Nibbana
  


I haven't found a more canonical source for the quote i.e. I don't know where that chant comes from.

The page Reflections on sharing blessings includes the Pali if that's helpful:


  Puññāni pakatāni me
  May they soon attain
  
  Sukhañca tividhaṃ dentu
  The threefold bliss
  
  Khippaṃ pāpetha vomataṃ
  And realize the Deathless.


The word translated as "bliss" is sukha, defined here (Wikipedia) and here (PTS dictionary).

The PTS dictionary definition includes:


  Two kinds, viz kāyika &amp; cetasika; at Pts.i.188; several other pairs at AN.i.80; three (praise, wealth, heaven) Iti.67; another three (manussa˚, dibba˚, nibbāna˚) Dhp-a.iii.51; four (possessing, making good use of possessions, having no debts, living a blameless life) AN.ii.69


Those three ("manussa˚, dibba˚, nibbāna˚") correspond to the three identified above (i.e. "of man, of gods, and nibbana") -- though I don't understand the reference, i.e. what "Dhp-a.iii.51" means.