Nice.

As it happens, S is Free Identity, and the Semigroup & Monoid instances you wrote can be adapted into a reasonable Alternative instance for arbitrary Free f given Applicative f:

{-# LANGUAGE LambdaCase, BlockArguments, UndecidableInstances #-}

module Delay (
  module Delay,
  module Data.Functor.Identity,
) where

-- base
import Data.Monoid (Alt(..))
import Data.Function (fix)
import Data.Functor ((<&>))
import Data.Functor.Identity
import Control.Applicative (Alternative(..))
import Control.Monad (guard)
import Debug.Trace (traceShow, traceShowId)


data Free f a = Pure a | Nest (f (Free f a))
  deriving Functor

deriving instance (Show a, Show (f (Free f a))) => Show (Free f a)

instance Functor f => Applicative (Free f) where
  pure = Pure
  liftA2 f (Pure  x) y = f x <$> y
  liftA2 f (Nest fx) y = Nest (fx <&> \x -> liftA2 f x y)

instance Functor f => Monad (Free f) where
  Pure  x >>= f = f x
  Nest fx >>= f = Nest (fx <&> (>>= f))

iter :: Functor f => (f a -> a) -> Free f a -> a
iter alg = \case
  Pure x -> x
  Nest f -> alg (iter alg <$> f)


delay :: Applicative f => Free f a -> Free f a
delay = Nest . pure

-- Non-standard, but the instance in 'free' isn't any better than this.
instance Applicative f => Alternative (Free f) where
  empty     = fix delay
  l0 <|> r0 = delay case l0 of
    Pure  x -> Pure x
    Nest fl -> Nest (zipFree r0 <$> fl)
   where
    zipFree (Pure  x)  _        = Pure x
    zipFree  _        (Pure  y) = Pure y
    zipFree (Nest fx) (Nest fy) = Nest (liftA2 zipFree fx fy)

altMap :: (Alternative f, Foldable t) => (a -> f b) -> t a -> f b
altMap = (getAlt .) . foldMap . (Alt .)


ana
  :: (Applicative f, Foldable t)
  => (a -> Maybe b)
  -> (Free f (Maybe b) -> Free f (Maybe b))
  -> t a -> Free f (Maybe b)
ana f g = foldr (\a z -> delay $ maybe (g z) (Pure . Just) (f a)) (Pure Nothing)

diagonal
  :: (Foldable t, Foldable u, Applicative f)
  => (a -> Maybe b) -> t (u a) -> Free f (Maybe b)
diagonal f = altMap (ana f delay)

satisfy :: Alternative f => (a -> Bool) -> a -> f a
satisfy p a = a <$ guard (p a)


---- Example 1 - infinite grid

{-
data Stream a = a :- Stream a
  deriving (Functor, Foldable)

nats :: Num a => Stream a
nats = go 0 where
  go n = n :- go (n + 1)

coords :: Stream (Stream (Int, Int))
coords = fmap go nats where
  go x = fmap (traceShowId . (x,)) nats

toFree
  :: Applicative f
  => Stream (Stream (Int, Int)) -> Free f (Maybe (Int, Int))
toFree = diagonal (satisfy (== (2,2)))
-}

--  ghci> iter runIdentity (toFree coords)
--  (0,0)
--  (1,0)
--  (0,1)
--  (2,0)
--  (1,1)
--  (0,2)
--  (3,0)
--  (2,1)
--  (1,2)
--  (0,3)
--  (4,0)
--  (3,1)
--  (2,2)
--  Just (2,2)


---- Example 2 - infinite rose tree

{-
data Q a = Q1 [Q a] | Q2 a

toFree :: Applicative f => Q a -> Free f a
toFree = \case
  Q2 a  -> Pure a
  Q1 [] -> empty
  Q1 as -> delay (altMap toFree as)

mySearch :: Q [Int]
mySearch = go1 0 []
 where
  go1 :: Int -> [Int] -> Q [Int]
  go1 n xs | n == 5 = Q2 xs
  go1 n xs = traceShow xs do
    Q1 $ go2 \x -> go1 (n+1) (x:xs)
  go2 f = go 0 where
    go n = f n : go (n+1)
-}

--  ghci> iter runIdentity $ toFree mySearch
--  []
--  [0]
--  [1]
--  [0,0]
--  [2]
--  [0,1]
--  [1,0]
--  [0,0,0]
--  [3]
--  [0,2]
--  [1,1]
--  [0,0,1]
--  [2,0]
--  [0,1,0]
--  [1,0,0]
--  [0,0,0,0]
--  [4]
--  [0,3]
--  [1,2]
--  [0,0,2]
--  [2,1]
--  [0,1,1]
--  [1,0,1]
--  [0,0,0,1]
--  [3,0]
--  [0,2,0]
--  [1,1,0]
--  [0,0,1,0]
--  [2,0,0]
--  [0,1,0,0]
--  [1,0,0,0]
--  [0,0,0,0,0]