This is a really interesting question. I have been looking around a bit in the resources I have to hand, but nothing that I'm seeing really talks about hearing loss as a reason for a pension.

Let me back up a second though and address a misconception that you have: the idea that people would be discharged from "service" or duty because of an injury. In general, during the period I study, men were not a member of "the Navy" as a body, but were instead enlisted on individual ships for the duration of a cruise, at the end of which the men would be paid off and become private citizens again. Now it was quite usual for commissions to last for many years, and for successful captains to be able to attract men either back to the ships they served in or to new ships if the captain moved elsewhere, but there was no permanent body of sailors in "the Navy" in the way we think of it today -- there wasn't a national registry or anything like that, or even a permanent term of service until the 1830s; the only men permanently attached to the navy were commissioned officers. So someone with hearing loss wouldn't be discharged for that per se, inasmuch as they weren't discharged from service in any case.

(The caveat to the above is that there were some warrant officers more or less permanently attached to their ships -- the gunner, bosun and carpenter, and often the sailing master. But that's a different conversation.)

Now, as to hearing loss itself, we do see references to temporary hearing loss after battle, especially in battles that were particularly noisy or violent -- L'Orient exploding at the Nile is a good example -- and there's no doubt that repeated exposure to noise will lead to someone's hearing degrading or going away entirely.

What we know about injury in the period, though, is based on a couple of sources: first, archaeology, which will not show evidence of hearing loss; and second, the writings of the time, which tend to classify illness separately from disability. It was a constant source of worry for the Admiralty to find lodging for sick sailors, and they would often contract with local women for extra space in their homes or house sailors at taverns and/or public houses until they recovered (with other deleterious effects upon health). But those arrangements were explicitly for sickness. Sailors who were somehow maimed during battle, say by losing a limb or an eye, were eligible for payments from the Chatham Chest; but those were for disability, not illness. Hearing loss seems to fall in between as I'm not really finding a reference to disability payment for hearing loss.

There was, of course, the famous incident of Robert Jenkins and the Spanish slicing off his ear, that led to the War of Jenkins' Ear; but he was a merchant captain and had no eligibility to receive disability for his wound and in any case the damage was external.

Now, as to the point of how often sailors would hear loud noises, that could be quite varied. Ships of course fired guns for signaling and saluting, and in battle, but the practice allowance of powder for guns during this period was quite limited, and captains such as Philip Broke who wanted to practice gunnery often had to buy their own powder and shot with private funds. So there's not an average amount of exposure to hearing loss that we know of.

Sorry to have written all this and have to end with saying "we don't know."

In the UK, Records at Kew (National Archives) hold documents for sailors and their pensions from the 17th to the 20th century. Greenwich had a hospital/retirement home for sailors from the 18th century onwards, and houses some records, conveniently a short stroll form the National Maritime Museum, whch also has a lot of records.

From what I can recall of my higher education in things of maritime history, battle wounds/associated injuries account for surprisingly few injuries; the is because a ship engages in battle for very little of its functioning life. The most common injuries and ailments are everyday things; gut problems (terrible food and scurvy); skin conditions (lice, poor hygiene etc); hernia and strains (no mechanical lifting aids); and accidents from falling from aloft. We also know that amputation does not always result in automatic retirement; in times of necessity they could still be serving. So logically, hearing loss and tinnitus may not have been quite so widespread; after all, the guns are not fired everyday, and when they are it's not a all day event - unless you are in battle. And of course, given that you are away from land for months at a time, you're not going to waste your ammunition on, bar for regular, rather than frequent, practice. The lack of mention of hearing problems may imply this, or I could be completely wrong. I'm fairly certain that hearing loss and tinnitus are caused by constant, persitant sound levels above acceptable range - given the relative small size of ships it seems unlikely that only a few would be affected (if it was many, it would have higher mention), and that such a source of noise was present in the pre-industrial age.

Temporary loss is certainly highly likely, but its repetition, leading to permanent loss/damage, that is the key. It may be worth mentioning that an brief work colleague of mine who had been an army engineer had hearing loss; the loss was in the lower registers, resulting in his setting his ringtone to some horrific high pitched squeak that drove everyone else nuts - but it was that only kind of noise he could reliably hear, especailly at a distance. It may be that the bosun's whistle was the 18t/19th century equivalent, so that hearing loss in the lower registers was something to be coped with and worked around, given that a significant part of everday life was routine and ritual. But I'm no expert...

Some injuries resuted in discharge and a pension, but tinnitus and hearing loss is not one I have personally come across. Undoubtably it occured and sailors were discharged for it. It was likely regarded as largely untreatable, however this is another area for research.

The book 'Jack Tar' by Roy & Leslie Adkins may be of interest to you, as would the institutions mentioned in the first paragraph.

As a sidenote, 'cannon' aboard a ship of war are referred to as guns, regardless of size. All cannon are guns, but not all guns are cannon.

I know nothing about the musket things, but similar principle applies, I would imagine?

Follow-up question: what of soldiers in large companies, firing off muskets before ear protection was created?

Can anyone recommend a good book, preferably a first hand account (diary, auto biography etc) by a sailor during this era?

How high were the tinnitus rates among battleship crewmen, especially those who served in fleets engaged in major naval battles?

As a side note I know that WW1 artillery men stuffed their ears with cotton. Would this not have been available to an 18th century sailor preparing for engagement?