Hmm, while I suspect you would prefer a general answer; this sort of question is notoriously hard to be precise with given the ramifications of each person's development growth and style.
Speaking from personal experience, I'm not yet at the level where I can play in exclusively (or majority) FIDE events at the appropriate level. So if we scale the question to how many ECF events should a player play in each year to improve, I can share some experience.
Biographical and general ramble warning.
First off, I'm still improving - I'm not the sort of player who exploded on the scene as an undergraded junior then with coaching, grew up and graded up year after year. I'm more the sort who has very big mountain-climbing legs, but who takes a while to make those big steps. It isn't as apt an example; but when I had been woodpushing as a kid, I ambled along until the age of 16 with a grade no higher than 55

But I'm almost perversely proud of this now. In my first year of sixth form I began to think differently, thanks to Teach Yourself Better Chess (Hartston), and went straight up to 107 (j). I then spent a few years polishing and modifying the thoughts I had at that level, until the point where despite having the discerning eye to spot mistakes my opponents made from a positional standpoint, I couldn't exploit them tactically. I hovered and hovered and got a job, and so my time for the game diminished - but then I changed employer and started working focusedly on my tactical prowess. Aged 24 I broke the old 120 barrier, but also had learned enough about myself (a process I am still engaged in deeply, partly due to diagnosis of Anxiety Disorder, which is actually a good thing from the absolutive standpoint) to start to maximise my potential in some games. I got to 136 but in the second half of 2010 I went on a tear and was performing at easily over 150. At that point though I saw an upper ceiling of ability based on the way I was playing - very calculatory, and tactically-based. I decided then and there to strike while the temporal iron was hot and totally build from the basics again from a positional/masterthought standpoint, even if it meant sacrificing my gains of the time. This appears to have worked, and I'm very proud to say I should be pushing 160 in the new grading list for summer.
How did I do this?
A lot of it has come from being able to be ruthlessly, Ronnie O'Sullivanly, hyperperfectionist about faults, and egoless in admitting them (though the way I write this, I come across as far from egoless!! :/ ). I played my first FIDE event in 2010, and since then have attended the British yearly - this summer holiday is equivalent to playing 5 rated events in one go, and I also play in the county championship here. While we could say "Dan plays in 6 ECF rated events a year", this could be misleading as a statistic to the original question. What if I played an event where in each game, I got 15 moves deep in the opening while using zero time, and had a healthy advantage in each game with neither any tactical melees or deep positional problems? On the contrary; what if I played an event where I got ground down to dust in each game without a clue what happened?
Thankfully for me neither of these has yet come about, and I remain for the most part able to extract the thread of play from games. Some players, while able to see where they went wrong in an individual game (antipositional idea, naive tactic etc), won't be able to track the patterns of their sin throughout the season, or, more tellingly, won't have access/time to working on the solutions to these. I'm lucky really: I work part-time, and I'm young, and, as if that wasn't enough, we have a strikingly competitive club championship, so I can't really say that I'm on a lonely plateau of understanding with people above and people below.
Given this, I would imagine that to the player's own content, the more games played, the better; with the proviso that it's better to play the sorts of event that suit your nature, but to challenge yourself too. It's no good going all-in for a weekender if you can't summon the heavy energy required of you to battle at a good level for the whole five games. Likewise, if you find it difficult to be an intense individual, you may be unsatisfied with 9 rounder FIDE events where you only have to crescendo once a day. I guess this is part of the beauty of playing a full British tournaments schedule for me: I'm stretching my comfort zone by forcing myself to pace my play (except in go for broke 5th round banzais), and the sheer volume in time of the games ensures that whatever my thought-form and general ability at the time, I'll immediately have games I'm capable of annotating/learning from deeply, rather than having to be in a particular mood like you might be if you play an exhausting league game that would require an equally taxing session in your study.
Looking at the grade histories of many of Britain's grandmasters, many of them played like the dickens in their young youth - Nosher is known for having played hundreds of games a year when very young, and the internet can only exaggerate this phenomenon now. Of course those jealousy-inducing smart bods have an exceptional ability to see and extract the data/information from their games, so copying them by trying to play 300 rated games a year will leave you little joy. Don't go playing hours of blitz if you can't perform at close to your best. If you're losing a piece or pawn to misvision more than once every 6-7 games, that's probably a red flag. (Says the man who was involved in two separate six hour games last summer alone...)
I guess it also depends on what improve means in your context. It might mean 'being able to more frequently play to your highest potential, given current knowledge', or it might mean 'gaining an attitude and desire to increase the volume of applicable knowledge within games', or subsets of this - a casual improving player might do some endgames, a junior might bonk his tactics rating first and worry about all else later. For me, improvement means something tangible. I get a real sense of moving up the monkey bars when I play a game and can self-reference an earlier effort where I did something that I now understand to be positionally suspect, AND am now able to go in the right direction instead (since it benefits nobody if you know that out of 4 ways to travel, south leads to death by crocodile, but you don't know which way doesn't). Or to put it another way; sometimes in a game I may find myself moving fluently in the sort of position where earlier I'd break down. That's a real buzz.