I've been a good customer with Discover for a number of years. Over that period, they've increased my limit and I've always paid on time, in full. Is it unreasonable to request a higher cash back rate to make their card more competitive with other cards? For example, I have the Amazon Visa card which gives me 2% and 3% cash back on gas stations and Amazon purchases respectively. Is it likely that Discover would increase my cash back rate for specific categories (say, gasoline purchases) to better compete with the other lenders I use?
The only way to change your cashback rewards is to change your card product type.
Discover's cashback rates are determined by the card product you're using. Discover's 2 main consumer cards are the Discover It card and the Discover Now card. Cashback rates are fixed on each card product because the merchant fees are fixed by card product and POS entry method (This is true for other card networks, too).
For example: If you do a swipe and sign (card present) transaction, the fee for the merchant might be different than that same merchant taking your card number over the phone (card not present) for the transaction.
You would be more likely to ask for additional "points" for being a loyal customer, unconnected to an individual transaction.
I have never heard of a custom cash-back rate before, but I have heard of people calling and asking for (and getting) an extra 10,000 miles because of their loyalty/spending/etc. This will vary based on card, the representative you get, and your card history.
You could always call them and ask. The worst they can do is say "no". It's not likely they're going to send people to your house to beat you up for daring to ask. It would be just as easy to call Discover and ask as to ask here.
That said, I'm guessing the answer is "no". Big companies like Discover aren't likely to make a custom credit card agreement for one customer -- not unless you're a billionaire, anyway. The only chance I'd see is if they have a different class of card that they regularly offer to other people, and if you call and ask they'll offer one to you.