Most giant tsunamis begin with seafloor displacement
Large shallow earthquakes at subduction zones can move huge areas of the seabed and launch waves across entire ocean basins. In deep water those waves can be low but extremely long and fast.
Coasts amplify the danger
As tsunami waves enter shallower water they slow and grow. Local bathymetry, bays, reefs and coastal topography can focus energy, which is why neighbouring stretches of coast can experience very different inundation.
Landslides can make extraordinary local run-up
In 1958 a rockfall into Alaska’s Lituya Bay generated a wave that stripped vegetation up to about 525 metres above sea level on the opposite slope. NOAA describes it as the highest recorded tsunami run-up, but its extreme height was local and decayed rapidly outside the bay.
There is no single universal maximum
The upper bound depends on the size, speed and geometry of the source and the shape of the receiving basin. Asteroid impacts or enormous slope failures can exceed ordinary earthquake-tsunami scales locally, but claims need to specify the physical source and geometry.
Research record
NOAA NCEI — Global Historical Tsunami DataOfficial · open sourceNOAA — Tsunami generation by landslidesOfficial · open sourceNOAA — The Tsunami StoryOfficial · open sourceHow Big Can a Tsunami Get?
Direct answer: Tsunamis can reach tens of metres at exposed coasts, while rare landslide-driven local events can run hundreds of metres up nearby terrain. A 500-metre local run-up is not the same as a 500-metre ocean-crossing wave.
Extreme local megatsunamis are real. Popular descriptions often inflate them by confusing run-up with offshore wave height.