Hi I recently purchased a cutting of this and would like to let it grow wild
to thinken. If I do so will it back bud when I eventually cut it back.

Do you have a standard Olea europaea or a dwarf cultivar like "Little Ollie?"
From my experience with "Little Ollie," it buds back very readily. The trouble
is, this cultivar is patented, so, like Ficus 'TooLittle,' it is usually sold
with a wink instead of a label.

Iris,
Central NY, Zone 5a, Sunset Zone 40
"The trouble with people is not that they don't know but that they know so much
that ain't so."
Josh Billings (Henry Wheeler Shaw), 1818-1885