The cogrowth series of a finite presentation is the generating function of all words in the generators that are equivalent to the identity. A result of Kouksov tells us that the cogrowth series is rational if and only the group is finite. In light of that it is natural to ask ``When is the cogrowth series algebraic?'' It is conjectured that the cogrowth series is algebraic if and only if the group is virtually-free.
We give three infinite families of group presentations having non-algebraic cogrowth series. These groups are close to but not virtually-free and so support this conjecture.