I was just wondering.
Would a transaction with a $0.01 fee get confirmed? If so, how long would it take?
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This changes with time, a lot. About 3/4 of a year ago, I made several test transactions spaced out over several days with very low fees, today worth between 1 and 5 ¢. (Unless stated differently: ¢ = 0.01 $; $ = USD) They all got confirmed within a few blocks time.
I just checked my transaction history and the one with the lowest fee I was able to find had a fee of 0.000023 BTC and a size of 226 B.
Right now, 1 BTC is worth 1'037.71 $. This means that my lowest fee of 0.000023 BTC is worth 2.39 ¢. However, it was worth less than 1 ¢ at the time.
Concrete info about the current state of affairs would get outdated very quickly. If you visit a page like http://bitcoinexchangerate.org/fees, it tells you the recommended fees for getting a transaction confirmed within different periods of time.
Right now, a transaction having a fee equivalent to only 1 ¢ will not get confirmed within any reasonable period of time because there are many unconfirmed transactions having a better fee per size than any transaction with only a fee equivalent to 1 ¢.
To give you a rough impression: The 48'000 transaction blockchain.info currently has in its mempool (go check https://blockchain.info/unconfirmed-transactions for the current numbers when you read this) have a cumulative fee of 54 BTC.
The last few blocks were all full (very common, right now) and collected an average of 1.66 BTC in fees. Go click the block heights in the top table of https://blockchain.info and find the average of the fees the last few blocks took. The table corresponding to each block has an entry called "Transaction Fees".
This means that 1 B in a block is worth 1.66 μBTC (if we neglect the block header). Even for a very small transaction of 226 B, that means it takes 226 B * 1.66 μBTC/B = 0.37516 mBTC to buy enough space for it in the blockchain. At Bitcoin's current value, thats equivalent to 0.3893 $.
Of course, the fact that some transactions which get in a block have a higher fee per size than others mean that you don't have to beat the average. But you have to get pretty close to it. And it's not looking like the BTC equivalence of 0.01 ¢ will (for any transactions) be pretty close to that average any time soon.
There will have to be a solution to so few transactions going through, compared to how many there are, though. There are several proposals to this and if implemented, those solutions would bring the cost required to get a transaction confirmed down.