How about tidal locking?
A planet close to a weaker star would not spin on its axis. Rather, it would point the same side at its star. Land on that side would broil. The dark side would freeze. On the middle band (the most habitable area) weather could blow in from either side swinging temperatures enormously. The interactions of these temperature extremes would drive extreme weather. Tornados could be a weekly fact of life.
Similarly, a tidally locked world with a big ocean facing the sun could develop very energetic ocean currents. An enormous and permanent hurricane might exist at the center of the sun-side, spinning off child hurricanes regularly.
A very big, dry planet
On Earth, the largest seasonal variations are usually found at the center of the largest land masses — think Siberia, central Asia or North Dakota. Proximity to large expanses of water tends to even out temperature extremes day to day and season to season.
Imagine a very large, earth-like world with maybe 30% ocean coverage. Land areas equal to the total surface of our Earth would have vast interiors far from large bodies of water. Seasonal temperature variations would be huge as would those of passing weather systems.
A geologically hyper-active planet
Large volcanic eruptions can change global temperature by filling the upper atmosphere with sun-blocking ash. A geologically hyper-active world could experience frequent mega-eruptions and subsequent nuclear winters. The timing of these would not be seasonal, but unpredictable.
Or maybe just a fast rotating planet
A planet turning very quickly on its axis would put a lot of energy into the weather system.