With the Cities expansion it goes up to eight too. Pretty simple to explain and as turns are simultaneous the extra player doesn't add any time (unless they are particularly slow)
Quest, which is an evolution of Avalon. It's a quicker game, because the voting phase has been eliminated. Instead, if the evil team win, the good team have a final chance to find the bad people. If you do play this game, I highly recomment playing with the "Director's Cut" rules. It can take 4-10 players.
Libertalia can go up to 6 players, and it's one of my favourites. I slightly prefer the new Winds of Galecrest version from Stonemaier Games, but any version of this game is fine.
Another game I'm on the lookout for is Feed The Kraken, which seems to be an improved version of Secret Hitler, but with a pirate ship.
I prefer Bang! The dice game over the regular Bang!, because it eliminates the early game where nobody shoots. There's almost always someone getting shot in every round even if you have no clue about the roles yet.
Ethnos - area control/ set collection/ push your luck with very short turns; hard to get to the table with my group because of the art, but once we play, everyone liked it so far
A good question and an interesting thread. At five, our group sometimes tries a substantial game like Viticulture or Power Grid. Knowing the game is critical, however, since many complex games are not at their best with five players. When our group hits six, we've recently been playing Mysterium, which is an excellent, simple investigation game with evocative art and a great deal of replay value. It's surprisingly accessible for new players.
With even larger groups, we sometimes play Mascarade. This Bruno Faidutti social deduction game is similar to his later Citadels, but easier to get started with. Another hit has been Bohnanza, although its wheeling-and-dealing style may not appeal to quieter folks.
When we have players who are easily overwhelmed by lots of rules, we tend to fall back to Sushi Go. This is entertaining enough--though beware of playing it with hate-drafting Magic players.
I love draft games (Magic player here, but I don't hate draft a ton). Been thinking about picking up Sushi Go but my current living situation severely limits my ability to play board games.
My go to would have been Deception: Murder in Hong Kong which I think needs between 6 and 9 players to really shine, but discounting social deduction, I'd look at Captain Sonar. The downside there is the player count is brittle... I do agree with @Zipheir that Bohnanza works well from 4-7. If 6 is acceptable, Point Salad is fine and that means you don't have to sort cards as you use them all.
Really, after 6 people, we break up into two groups. I mean, sometimes we break up at 6...
Cash n Guns - 8 players:
Players get to have a standoff with foam guns pointed at one another trying to take out each other either with a loaded gun or bluffing that your gun is loaded to gain the most loot that was gathered from a prior heist.
We default to Munchkin when nothing else seems obvious. Games get long with 7 or more people because someone can always sabotage the leaders victory. But the game is still hilarious and gleefully fun to us after all these years.
It's a Wonderful World (up to 7 players with the expansion )a kind of 7 Wonders with resources and combos. Actually, we don't play 7 Wonders anymore.
Condordia Venus (up to 6 players) very euro-style, move your workers on the map to build warehouses to gather resources to build more stuff and buy more actions..
Root (up to 6 players) very asymetric strategy game. It's so asymetric that each faction plays a different game.
Demeter and Cartographer (almost illimited number of players) like tons of other roll and write, they are the perfect games when there are too much players for other games and you don't want to play a party game.
Also, lots of games are fine if you add extra players with a few tweaks: Mansions of Madness, Pandemic, Res Arcana..
I've had four player meets turn into five players a few times. I'd say Hansa Teutonica does great in that situation. It's a game that doesn't do as well at lower player counts anyway, so it all works out.
6 players is usually the threshold where I split into two games, so 5 is usually the mex I look for a game to support. Others have listed some great games in this slot.
One more obscure one Ill add is Veiled Fate. Great twist on thehidden role game, and supports up to 8 players.
I think that Keyflower scales really well if you like Euro's. I have played it at all counts and I tend to prefer it with more people. It can be a really mean game in some ways (everything is an auction essentially), but there is a nice spacial element, and a lot of strategy. It also doesn't take too long, and the rules are very straightforward and well presented in the manual.
If you have the time for it and prefer the opposite then Dune (the reprint of the 1978 game) is best with 6. It is a very social game, nice strategy, and not too fiddly, also can be very mean.
Shadow Hunters is one of my favorites for 6-8 players, it's not out of print but I assume the reprint "Fangs" is similar. Truro is a good simple game, and it stays quick from 1-8 players
Scythe has an expansion that goes to 6 or 7, I think. Some versions of Smallworld may as well. Formula D plays 10. Dixit can go high too if you have enough cards.
Mysterium - identify the suspect by looking at abstract cards given by a ghost (goes up to 7 players I think; use the original Ukrainian rules!) Shikoku - interesting movement game, goes up to eight players Eclipse - epic space exploration and battles, nine players with the expansion (I've never played it nine players, it would probably take about a week and need a banqueting table to put it on)
I agree about Werewolf. It's especially nice that the set of characters can be varied for lesser or more experienced groups.
I've had a terribly boring time with large-group (five or more) Wingspan. Admittedly, I'm not very enthusiastic about the mechanics in the first place, but the game seems to drag badly with more than four players. I also would not recommend it for groups with inexperienced players. Wingspan's combo-oriented engine-building can be punishing for those without "system mastery", as can the odd "score n points for each bird with a red tail who's looking to the left" objectives.
Big fan of The Resistance, as well as Blood on the Clocktower, though the latter is a bit of an investment. If you don't want social deduction, there's also 7 Wonders and Citadels.