I did not ‘give my word’, I entered into a business contract, governed by state law. Big difference. I feel absolutely no ethical requirement to waive my rights under the law.
My only real concern would be someone trying to claim an old debt long after I’ve forgotten the particulars. That’s why I suggested diarizing it for future reference (I think FairyChatMom and I are close in age, and if her brain is doing what mine is, she’ll need the details written down). I can see it happening even years down the road; a new accountant gets hold of old books, and goes to town trying to collect all the old invoices.
Let’s not pick on my memory now… 
After all, I still remember that he hasn’t billed us.
And frankly, I expect he’s forgotten because he didn’t write anything down, especially the part where we ended up staying out of the water for several days. So I don’t expect an audit to show us as an old debt.
We tried running our own business several times, and I know every penny made a difference. And while we never had cash flow on the same level as the marina, (between slip rents, haulout fees, fuel sales, retail sales, transient boats… ) our few hundred dollar debt is a tiny percentage, but still, it bugs me.
On the other hand, he’s running a business - he has a certain responsibility here, too.
Fro this amount, I think I’d send an email (or failing that, a letter–which in this case suffers by being overly formal) saying I was trying to balance my books for that month, and could he go ahead and send a bill? Ideally I’d like a bill by the end of the month so I could get everything figured out in time for Christmas shopping season. Make it light and chatty, but with a requested deadline.
It sounds like the OP doesn’t know the amount of the bill. The boatyard has an obligation to send a bill with the specific amount if they expect payment in reasonable time. I’d ask once to find out what the amount was. If I had no way to make a reasonable estimate, it would make sense to keep asking to find out what someone would want to collect some day, or if I wanted to dispute the amount.
On the other hand, based on an actual incident at a restaurant, I waited for 30 minutes to get a bill, went to the front to try to find a host or manager, waited 15 minutes after being told someone would be with me in a moment, and then left when no one showed up. Someone came running out to the parking lot, so I went back in and paid. I think I would have just left and never returned to pay after that treatment if nobody came out. But I was young then, and in retrospect, I would have been obligated to pay them somehow.
I would not put that in writing, you want to keep it on a personal light level, not pushy somewhat adversarial level. If that pushy level happens it will cause a feeling of uncomfortableness, which you don’t want.
But then again you may feel this already, waiting for the ‘shoe to drop’ so to speak, which is not good either.
You asked three times? I’d forget about it, but pay it if/when he decides to get around to billing.
It may be that this is his busy season and once things calm down he’ll get around to the billing part.