I simply want to buy $40 worth of bitcoin and my question is: Is there anyway for me to do so without incurring a ridiculous fee for the transaction?
Yes.
Isn't there usually a minimum fee of $15 for buying bitcoin no matter what the amount?
That depends on where you are buying, but generally no. Most of the sellers I know of sell for a fee of 0% to 10%. That would be a maximum fee of $4 to buy $40 worth, and possibly less.
Would it be possible to purchase $40 dollars worth of bitcoin and only pay the standard 1% fee for this?
Why do you say 1% is a "standard" fee? You
might be able to buy with a 1% fee, you will
probably be able to buy with a 5% fee, and you will certainly be able to buy with a 10% fee.
Can anyone provide me with some guidance on this and if possible links to a bitcoin exchange where this type of transaction would be possible.
https://localbitcoins.com/Depending on how much of a hurry you are in, and where you are located, I'd be happy to sell you some bitcoins.
If you are anywhere near the Greater Chicagoland Area, we could meet up so you could give me the cash.
If you are not, then you could put the cash (or a USPS money order) in an envelope and mail it to me.