You can make a game work with anywhere from 100 to 5,000 chips. The question is what trade-offs are you making when using more or fewer chips?
You need enough high-denomination chips to make a bank big enough to cover, as
@allforcharity says, "all the money on the table on your biggest night". How much that is depends on your stakes, your buy-in and rebuy limits, the games you play, and how aggressive and spendy your players are. A typical amount might be three buy-ins per player with 100bb buy-ins but in practice your game might need only a fraction of that or might need several multiples of that.
If you don't have enough big-denom chips to cover your bank, then you can still make do by letting cash play. If you only have quarters, ones, and fives, but are expecting multiple hundred-dollar buy-ins, what do you do? Let people put $20 bills on their stack. Problem solved.
You need enough low-denomination chips to let people place their bets conveniently using chips already in their stack rather than needing to get change from other players or from the pot. There's no magic number for this, though, because once again it depends on the betting habits of your players. Fewer chips means more change-making, but change-making is hardly the end of the world, so if your players use fewer chips when they bet and don't have any difficulty or annoyance making change then you can get away with having fewer chips on the table.
Then there's the question of giving out low-denoms or high-denoms when handing out add-ons and rebuys. If someone busts out and rebuys, you could hand them a few high-denom chips and let them make change from the other players at the table. But there's something aesthetically pleasing about letting players who took someone else's stack
keep all the chips they just won. If you have enough low denoms, you can give the rebuying player a new stack that already has all the low-denom chips he'll need for making bets, which means no change-making, which means the low-denom chips he just lost get to stay in the stacks of the players who won them. This also means that over time the stacks at the table get
physically larger, which is cool. The alternative is that the stacks
accumulate a few high-value chips (or bills if cash plays)
, which is also appealing but is a different aesthetic from having larger piles of chips.
With fewer low-denom chips in play, there will be more change-making and stacks will be less able to grow physically large over time. But again, the point is that there's no magic number which is the right amount. It's a continuum.
Vintage poker chip sets like the carousel you found typically came with room for 200 chips, with a pre-determined breakdown of 100 whites, 50 reds, and 50 blues. Here's how you could use that breakdown to support a modern game of 25c/50c NLHE:
First off: make blues 25c, whites $1, and reds $5. In olden days the blue chip would be the highest-value chip, but NLHE games today need more of the second denom and fewer of the first denom, and the old breakdowns aren't designed for that. So swap them around to make it work better.
Assume buy-ins are $50, which is 100bb.
The first four players each get:
12 blue (12x 25c = $3)
17 white ($17)
6 red (6x $5 = $30)
The fifth player gets:
20 white
6 red
The sixth and seventh player get 10 red each. That puts all the chips in play except two blue and twelve white.
Subsequent players plus any rebuys or add-ons put twenties in cash on the table.
This will work pretty well for four or five players without a lot of change-making. With six you start to make change more often, and it gets even more frequent with seven, eight, or nine players. At ten players the low-denoms are spread super-thin so change-making is simply a constant activity. It might get on your players' nerves, but it's still
playable.
The carousel you bought has a 300 chip capacity, and you're not constrained to using just three denoms. That means you have a great deal of flexibility in how to stock it with chips. Depending on your stakes, your buy-in and rebuy limits, the games you play, and how your players like to bet, you may want to have more or fewer low-denoms for betting with and more or fewer high-denoms to make a large bank. Nearly any breakdown will
work, but what works
best is something you'll have to find out for yourself.