The Monero client allows configuring the blockchain location. My assumption is that only public blockchain information gets stored here, so I configure it to an unencrypted SSD.
However, it seems possible that the client would cache other information, like which transactions or unspent outputs are mine, in order to resume syncing quickly after a restart.
Besides the public blockchain, what else does the client write, and where? Does any of it reveal information about my address or transactions?
As you surmised, monerod stores the blockchain, which is public information. It can reside on an unencrypted partition if necessary. The daemon does not have access to any wallet secret keys. The blockchain is stored in $HOME/.bitmonero/lmdb. Other ancillary files stored by the daemon are the log files and the set of known peers, both in $HOME/.bitmonero.
monero-wallet-cli does have access to the keys, and stores three files: NAME.keys (encrypted with your password, contains the secret keys), NAME (the wallet cache, encrypted with your secret key from the first file), and NAME.address.txt (plaintext, your standard address). These files are stored wherever you decide to create them.
The cache stores all the information which monero-wallet-cli parsed from the blockchain which is related to that wallet: incoming and outgoing transactions, key images, tx secret keys, address book, etc. If this cache file is deleted, it will be recreated by parsing the blockchain again, but you will lose what's not saved in it, such as tx secret keys, address book...
The address file is unused after creation, so you can delete it if you wish.
The Monero daemon (started by hand or automatically by monero-wallet-gui) only writes data that is publicly available (the blockchain). Therefore you can choose a directory on an unencrypted partition as data directory for monerod.
The Monero wallet (monero-wallet-cli, monero-wallet-rpc or monero-wallet-gui) writes the wallet files, which must stay private. These files are encrypted using your password, so they can also be stored on an unencrypted partition. But storing the wallet on an encrypted partition will make it even more difficult for someone stealing your computer to try to steal your money.
