Chlorine bleach is the first one that comes to mind. If an acid like vinegar is added to it, it produces poisonous and irritating gases such as chlorine and hydrogen chloride (which makes hydrochloric acid when it contacts the water in your eyes, throat, and nose).
Mixing bleach with ammonia can produce poisonous chloramine vapors.
Most of the strong chemicals in the house should not be mixed, if only because it makes them less effective. But when strong chemicals are mixed together, they often generate a lot of heat, and can boil, splattering caustic chemicals onto your skin or into your eyes. Lye and drain cleaners have to be very carefully handled for this reason. They can boil the water they are added to.
Pesticides should never be mixed with anything other than what the label recommends for diluting them. They are poisons to start with, and causing a chemical reaction can liberate poisonous vapors or gases that you might not be able to smell, so you might not know you are poisoning yourself.