High is a relative term. How high is high? One example is diamond changing to either graphite or amorphous carbon. Diamond is not the stable form of carbon but the activation energy is so high that it doesn't change to carbon very fast. Another example is a match. Matches don't burn by themselves; they require something to get them started. The same can be said for natural gas, propane, etc. Even the reaction between hydrogen and oxygen to prepare water needs a spark to get the reaction started. H2 gas and O2 gas mixed together will not react but the spark that sets them going will cause an explosion. Take your pick. There are more than two here but they require a range of activation energies. Take your pick of what you think is high.
Thank you for the examples given. The teacher treats the gasoline explosion as low energy activation example because the gasoline only requires a spark to ignite. Do you agree it is a low energy activation example? By your example, I think that rusty nails is a high energy activation example. Agree?
I agree that the explosion of gasoline is a low energy of activation as is H2 and O2 explosion with a spark. I would agree that the rusting of a nail is a higher AE than gasoline explosion but, to repeat myself, high is a relative term. And I'm not so sure that it is absolutely correct to correlate the extent of the reaction or the speed of a reaction with the height of the AE. For example, I could use an atom bomb to light a match but that wouldn't tell me much about how high the AE was for the match.