【Addition】 , 【accretion】 , 【increment】 and  【accession】 all agree in denoting a thing that serves to increase another in size, amount, or content.

【Addition】 implies union with something already existing as a whole or as a unit.

• he built an 【addition】 to his house last year
• the office boy, a recent 【addition】 to the staff, was busy with the copying press
  —Archibald Marshall

Sometimes improvement rather than increase is stressed.

• the paintings were an 【addition】 to the room

【Accretion】 implies attachment from the outside; it may be used of the process as well as of the thing added.

• a rolled snowball grows by 【accretion】

It often suggests additions made to an original body over a considerable period of time.

• the professional historian, whose aim is exact truth, should brush aside the glittering accretions of fiction that have encrusted it
  —Grandgent

Nearly always it implies the 【addition】 of unessential or alien matter.

• all progress in literary style lies in the heroic resolve to cast aside accretions and exuberances
  —Ellis

【Increment】 usually implies 【addition】 bit by bit in consecutive or serial order.

• the salaries are raised by annual increments
• one more wave in the endless ebb and flow of action and reaction, the infinitesimal increments of which we call Progress
  —Lowes

Sometimes it signifies increase in value.

• benefited from an unearned 【increment】 in the value of his land resulting from growth of the city

【Accession】 denotes something acquired that constitutes an 【addition】 to contents, holdings, or possessions.

• recent accessions to a library
• the greatest 【accession】 of positive knowledge has come in our own time
  —Inge