So if Legendary Monsters aren't meant to be Solos (rather than simply being as badly designed as MM1 solos) then there is no way out of the box that a party in a game of Dungeons and Dragons should have a decent fight against a dragon without a pack of minions. Is Dungeons and Dragons really intended to lack showdowns with the party vs a dragon?
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I think the old D&D mind set that a dragon may be nearly invincible is definitely turned on its head with 5e, but here is what 5e offers:
A party of 4 or 5 5th level PCs can encounter an adult dragon, get hurt pretty badly (but do enough damage to make the dragon flee). Then, when the group encounters it again at 7th or 8th level, they can get revenge.
For an ancient dragon I'm not sure how the numbers work, but maybe it's more like 8th level you get your butts handed to you, but 12th level you can take it. (I haven't run one this high yet so these numbers are just made up).
I'm totally cool with letting DM modify dragons to suit his or her campaign. Just giving a dragon Heavy Armor Mastery (some kind of DR), or Sentinel feat, or resistance vs bludgeoning, slashing, piercing, can go a long way to making it stand out. Also, spellcasting dragons can be a real killer especially if they can cast fireballs or lightning bolt or other big damage area spells.
A lot of a fight with a dragon might depend on whether or not it recharges its breath weapon too. If it can use its breath, then fight for a round or two using spells or just its normal weaponry, it may get another chance to breath (every 3rd round on average).