The occasional exceptions drive my players crazy. A CR 3 sabre-tooth tiger mummy on the Isle of Dread (Goodman Games version) inflicted mummy rot on a PC last session. The party is 4th level, and there wasn’t an NPC caster for dozens of miles around. Without access to
remove curse...there is literally nothing you can do. So the PC crumbled to dust 48 hours later. Now to get the character back it will take at least a
reincarnation (if they can somehow reach the Druid in 10 days and he happens to have the components for it) or
resurrection spell at some indefinite point in the future when they come across a helpful high level cleric.
(The PC came back as a ghost, because that’s the kind of stuff that happens in this campaign for reasons the PCs don’t know, but they still found the unavoidable deadliness of the CR 3 critter messed up. Kids these days!

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Absolutely 100% agree that curses aren't that interesting in D&D. Either you break them with
remove curse (or similar magic) or you suffer the effect. It's very binary and doesn't require any thinking – of course they're going to break the curse if that becomes an option. That's nothing unique to 5e. Pretty much curses have always been like that in D&D.
It's possible for a DM to make an interesting curse from scratch with clever wording, but it does require some bending over backwards.
I often house rule that a curse must have an "out" clause, often phrased as
"You shall be cursed with ----- , until -----" or
"Every ----- , you shall be cursed with -----, until -----" or
"By the light of -----, you shall be cursed with -----, until -----". Fulfilling the condition (which is rarely easy or 100% obvious) lifts the curse, causing it to stop actively affecting the character. However, curses often include clauses that cause the curse to recur or be re-cast. The sun sets. You kill another creature. You meet a one-eyed man. Whatever. To permanently end a curse requires fulfilling the condition, followed by a casting of
remove curse. Another way to think about this is that I house rule a special component for
remove curse – whatever clause is in the curse.
What this does is shift the binary (i.e. supremely easy to break / save or die) nature of being cursed to a
save... and then die paradigm. This allows all players to applying their creative thinking to figure out how to lift a curse temporarily. Maybe you can't figure it out, and then you suffer its effects, but at least there was the feeling that you had a chance to puzzle your way out of danger.
Remove curse becomes the cherry on the cake to break the curse so that you don't have to worry about repeating the condition that lifts it every day (or whatever time period the curse was cast to recur on).
This requires a bit more thinking about curses, and finding ways to tie them to the narrative so the presence of lift conditions is foreshadowed to the players, but so far we've found it more satisfying than the default treatment of curses.
Would have been great if there was a half page to a page devoted to curses in the DMG – similar to how there is for poisons & for madness – with some tables for creating playable curses.