You can transfer ether without a contract. This is used to be (and probably still normally is) the most common type of transaction on the Ethereum blockchain.
I'm not sure what you mean by "without using something Bitcoin doesn't have" -- do you mean without using smart contracts, can you do everything Bitcoin does at the protocol level? There are some things, like paying multiple addresses in one transaction and also using multiple accounts to fund a single transaction that require something more complicated than a standard transaction in Ethereum; these are pretty standard in Bitcoin.
In fact, bitcoin uses scripts for transactions. You can do things like have someone solve a puzzle to claim some bitcoin or you can perform a proof of burn. While less powerful than Ethereum contracts, your question asks about what can be done without Ethereum contracts (I think). Things like multi-signature wallets in bitcoin are another common-ish thing that can be done in bitcoin but require a contract in Ethereum.
Historically, the bitcoin protocol also included an alert feature which would allow someone possessing the secret alert key to broadcast a message to all bitcoin users. This system has been retired.