I know there is a word for when there is a picture inside a picture, that inside another and so on, recursively. Does anyone know this word?
It is the Droste effect.
The Droste effect - known as mise en abyme in heraldry - is the effect of a picture appearing within itself, in a place where a similar picture would realistically be expected to appear. The appearance is recursive: the smaller version contains an even smaller version of the picture, and so on. Only in theory could this go on forever; practically, it continues only as long as the resolution of the picture allows, which is relatively short, since each iteration geometrically reduces the picture's size. ... The effect is named after the image on the tins and boxes of Droste cocoa powder, one of the main Dutch brands, which displayed a nurse carrying a serving tray with a cup of hot chocolate and a box with the same image. This image, introduced in 1904 and maintained for decades with slight variations, became a household notion. Reportedly, poet and columnist Nico Scheepmaker introduced wider usage of the term in the late 1970s.
I would say that 'recursive' is the best word for this. I noticed that you did not use the word and that an editor changed your question to include it, so I want to make it clear that this is exactly the kind of thing that the word 'recursive' is used to describe.
The answer from JLG is interesting (and I voted it up), but it is also obscure and a name for the phenomenon rather than a word that describes the phenomenon.
Again, I think the edit made here was sufficiently heavy as to possibly conceal the aim and level of your question, so I wanted to provide an alternative response that I think may better suit what you originally asked.
There is a French expression made especially for that, I think English speaking people could use it : "Mise en abyme".