Are Bitcoin transactions permitted to have no outputs (i.e. all inputs become transaction fee)? I am interested in the possible use of the Bitcoin (or other altcoin) blockchain as a generalized distributed ledger, meaning that it is useful for tracking information other than coin transactions proper. A prerequisite for that would seem to be the ability to pay for transactions whose primary purpose is not moving Bitcoins from one address to another. So, can Bitcoin transactions list no outputs, thus leaving all inputs as a transaction fee reward for the miner?
No. If you create one, it won't be relayed or mined by Bitcoin Core. If it gets into a block, it will be rejected. From the source code for Bitcoin Core (tx_verify.cpp:164):

if (tx.vout.empty())
    return state.DoS(10, false, REJECT_INVALID, "bad-txns-vout-empty");


However, you can create a vout with 0 satoshis. That isn't a "standard" transaction, so it will not be mined by the software by default. However, if it gets into a block, it's valid.