The Lightning-RFC defines, in BOLT #11, an invoice protocol for proof-of-payment. While this is a pretty cool feature, I was wondering if it would be possible to make a payment without such invoice. That is, with the origin node defining a payment hash etc.
With the current specification it is not possible to make a spontaneous payment to somebody on the network. However, this will likely change very soon, as it is one of the features on the roadmap and will be discussed in the specification group in November.
There are tricks on how to do a payment by delivering a large fee at the destination and continue routing to someone else (described in this video), but you have to write software to manipulate the route and the destination will not see an incoming payment, since it considers it a fee.
Some wallets will let you create 0 sat invoices, which the payer can then specify the amount they want to send.