Rainbow Islands: "The game is sometimes called "Bubble Bobble 2". However, two other games --"Bubble Symphony: The Story of Bubble Bobble 2" and "Bubble Bobble 2"-- claim the same name. Rainbow Islands happens after Bubble Bobble. Bubble Symphony happens after Bubble Memories, which is called Bubble Bobble 3, so Bubble Symphony is really Bubble Bobble 4 or 5. Bubble Bobble 2 also happens after Bubble Memories, so it is either Bubble Bobble 4 or 5. Bubble Symphony's song titles, which refer to Bub and Bob as "Grandfather", are true, then Bubble Symphony is Bubble Bobble 5 and Bubble Bobble 2, which features Robby, the "son", is Bubble Bobble 4. But then again, the intro of Bubble Symphony refers to Bubby and Bobby as the children of Bub and Bob, so Bubble Symphony is Bubble Bobble 4".
Rainbow Islands is BB *Part* 2 - not BB2 - and Then there's RI the Extra Edition, which is BB part 3 - but still a part of BB 1, featuring Bub and Bob. I think the the intro of Symphony calling them "children" in the descendants sense, not the direct children sense. Like "Children of Israel" can refer to all of the Jewish people - not just Jacob's 12 sons and 1 daughter.
So - BB 1 includes BB, RI, and RIEE. BB3 is Bubble Memories. BB4 is Bubble Bobble 2. BB5 is Bubble Symphony. There is no BB2.