TITLE: Are monads monadic?
QUESTION [13 upvotes]: Is there some sort of monad whose algebras are monads? How about if we are internal to a bicategory B? Are internal monads in B monadic? Certainly not always, as otherwise free T-multicategories a la Leinster would always exist. What is known?

REPLY [10 votes]: An important reference on free monoids (and by extension, free monads) is:

G.M. Kelly, A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, Bulletin of the Australian Mathematical Society 22 (1980), 1--83.

It's 83 pages long and famously impenetrable.  But those who've braved it say that it contains just about every free monoid construction that anyone's ever thought of.  
So if you want to construct a free monoid/monad, the question is whether it's easier to (a) read Kelly's paper, or (b) do it yourself and cite Kelly anyway (on the assumption that your construction is probably in there, somewhere).