Surprisingly, there are several nine-letter words that can have one letter removed in each round to make a new word all the way to one letter, which must be ‘a’ or ‘i’. Startling is the most common answer, but I’ve included the other words I’m aware of. I don’t include plural words, like cleansers, drownings, splatters, starvings, trappings and wrappings because it’s kind of cheating.

startling
Remove the l to make starting (or remove the t to make starling)
Remove the t to make staring
Remove the a to make string
Remove the r to make sting
Remove the t to make sing
Remove the g to make sin
Remove the s to make in
Remove the n to make I

splitting
Remove the l to make spitting
Remove the p to make sitting
Remove a t to make siting
Remove the first i to make sting
Remove the s to make ting
Remove the g to make tin
Remove the t to make in
Remove the n to make I

stringier
Remove the r to make stingier
Remove the i to make stinger
Remove the t to make singer
Remove the r to make singe
Remove the g to make sine
Remove the e to make sin
Remove the s to make in
Remove the n to make I

strapping
Remove the s to make trapping
Remove the t to make rapping
Remove the p to make raping
Remove the r to make aping
Remove the a to make ping
Remove the g to make pin
Remove the p to make in
Remove the n to make I