Well no the lowest fee is 0 BTC (for high priority transactions). Only low priority transaction have a mandatory fee as a denial of service prevention mechanism. All tx fee (both mandatory or optional) go to miners.
Excuse my sounding like a newb, but this escaped me. What are low/high priority tx?
Notme posted the exact formula but a general rule of thumb is
"one bitcoin day" for high priority. By one bitcoin day I simply mean the value of the input (in full BTC) times its age (how long ago the coin was received). If your inputs are "one bitcoin day" or greater then your tx is high priority and no mandatory fee is required. You may still wish to include an optional fee to faster processing but the tx will be relayed by all nodes as having sufficient priority for no fee.
For example.
1 Bitcoin received 1 day ago (144 blocks) is high priority (1 BTC) * (1 day) >= 1 bitcoin day
0.5 Bitcoins received 2 days ago is high priority. (0.5 BTC) * (2 days) >= 1 bitcoin day
2 Bitcoins received 12 hours ago is high priority. (2 BTC) * (0.5 days) >= 1 bitcoin day
Times values in hours or days are just illustrative all age is by blocks 144 blocks = 1 day on the network (regardless of it being shorter or longer than one calender day).