I've heard this construction a few times.
What does it mean exactly, and is it grammatically correct?
For example:
"it underwent a number of changes until it got to being exactly what I wanted it to be."
"I got to thinking about old times"
It seems to me it's just about synonymous to get around to doing something, or (start doing something)
The problem with "it underwent a number of changes until it got to being exactly what I wanted it to be" is that the viewpoint changes three times.
This is a passive expression viewing the object as a passive agent without an active agent acting upon it. It is a bare statement of event which ascribes no activity to the object itself and ascribes no activity to any other agent.
This is somewhat ambiguous but could be understood to mean that the object had something to do with its 'getting' to the 'state of being' which resulted.
This changes the whole aspect of the concept and now ascribes total control to 'I'. It now becomes apparent that 'I' am the sole agent and am responsible for all that has gone before.
I think that, conceptually, the change of viewpoint is undesirable and this results in what was described in comment as an 'awkward/non-idiomatic' context.
It means slightly different things in each example.
In the first, it means 'reached'. It's a clunky sentence construction but means, 'It underwent a number of changes until it reached perfection'.
In the second, it means 'started'. 'I started thinking about old times.'
The expression, 'I got to thinking...' is generally used when something prompts you into reminiscing about something else. 'After seeing my grandmother, I got to thinking about all the times she baked me chicken pie.'
HTH
