This question will get more interesting responses if it was qualified as:
"Does anyone run Postgres without PgBouncer for non-trivial workloads?"
Because, as we can see from the comments so far, lots of people are going to say you don't need it for your blog that gets 10 hits a month.
I've personally never heard of anyone not using PGbouncer, or some connection pooling proxy, for reasonably concurrent workloads. PG's process-per-connection architecture almost requires it. Otherwise even a small connection storm will wreak havoc on your server.
I feel like there are a lot of use cases where I’d opt for SQLite and a lot of use cases where I’d opt for Postgres + PgBouncer. I’m curious what kinds of features push towards using Postgres alone over SQLite.
Good question.
PGBouncer isn't as useful if you have seriously long-running transactions. It can’t do much of anything with those. Sure, you can give it a pool of 1000 and your Postgres instance a pool of 100, but you’re just moving who is going to say, “sorry, the database can’t handle your request right now.”
It’s not a silver bullet.
If you have more connections than Postgres can handle on the hardware it’s on, but with a bit of buffer it’ll be able to burn them down: great.
If you have long-held connections with many transactions that PGBouncer can interleave: great.
If you have connections whose transactions are longer than a reasonable timeout, well, your optimization princess is in another castle.