Is it possible to send 0.00000001 BTC? I tried doing it using bitcoin-qt but it wouldn't let me. Is this to avoid blockchain spam?

What clients would allow me to send that amount?

Will I have to pay fee if I send such a small amount?
You can not send transactions that small. The minimum is 0.00005460 BTC, anything smaller is called dust and will not be accepted into the network. You will most likely need to pay a fee with something with this little priority, which will cost you more than the amount you're sending. Bitcoin is not intended for micro transactions. Also bear in mind that collecting a lot of transactions this small will cost you more in fees to spend them again than the value of the coins.