MySQL / PHP - Encuentra franjas horarias disponibles

SELECT a.end AS free_after
FROM bookings a
WHERE NOT EXISTS (
  SELECT 1
  FROM bookings b
  WHERE b.start BETWEEN a.end AND a.end + INTERVAL your_duration HOURS
)
AND a.end BETWEEN start_of_search_window AND end_of_search_window;

solo necesita proporcionar valores para your_duration (entero), start_of_search_window (fecha y hora) y end_of_search_window (fecha hora).

Y si quieres campanas y silbatos...

SELECT free_from, free_until
FROM (
  SELECT a.end AS free_from,
  (SELECT MIN(c.start)
   FROM bookings c
   WHERE c.start > a.end) as free_until
  FROM bookings a
  WHERE NOT EXISTS (
    SELECT 1
    FROM bookings b
    WHERE b.start BETWEEN a.end AND a.end + INTERVAL your_duration HOUR
  )
  AND a.end BETWEEN start_of_search_window AND end_of_search_window
)
ORDER BY free_until-free_from
LIMIT 0,3;

Le dará las tres ranuras disponibles más cortas (es decir, las más cercanas al tamaño de destino) dentro de la ventana.