I'm trying to create a BBEG that can't die naturally, or at least lives for thousands of years. The PCs need to be able to kill them in the end though. I know I can use a Lich or Vampire but is there any other way to accomplish this?

I'm not interested in monsters who are already immortal, I am interested in finding a mechanic that will allow my BBEG to be immortal regardless of their race. I am considering making BBEG a Loxodon, but have not yet settled on anything. You can assume that BBEG remains on the material plane for the duration.

The final battle will be at about Level 15 for the PCs.

Have them be a spellcaster with access to high level spells

The easiest way to achieve effective immortality is to use spells. A wizard has access to several of them once they get higher level spell slots.

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  Sequester will allow the BBEG to put themselves into a state of suspended animation during which they will not age. This will allow them to span any amount of time without aging (though it will mean that they are out of action for that time).


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  Clone will provide the BBEG with fresh young bodies to use once their older body dies. With enough of these, they can last an infinite amount of time. See this Q&A for more discussion on this.


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  Wish obviously wish can do anything, including granting immortality. A wish for immortality is unlikely to be granted in a straightforward manner though and would be a good opportunity for the DM to implement some sort of curse or other downside on the BBEG as a result of their wish.


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  Imprisonment is an option but it would require someone else to cast it on BBEG and for them to let them out again.


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  Reincarnate, revivify, resurrection, true resurrection can obviously also bring BBEG back from death assuming there is another caster capable and willing to cast it on them.

Be a high level Druid or Paladin

At 18th level, druids get the Timeless Body feature:

[T]he primal magic that you wield causes you to age more slowly. For every 10 years that pass, your body ages only 1 year.

Combine this with a long-lived race like the elf or warforged (see below) and you easily have thousands of years of life.

An Oath of Ancients paladin at level 15 gets:

Additionally, you suffer none of the drawbacks of old age, and you
can’t be aged magically.

Which seems likely to be able to prevent death from old age as well. (thanks @Walt)

Grant an Epic Boon

As Carcer points out in their answer, a DM can use the Epic Boon of Immortality to grant BBEG Immortality. This assumes that they have 20 PC class levels and may not be justification enough for BBEG being immortal plot-wise, but it is an available mechanic.

Have them be a warforged

Warforged, from Wayfinder's Guide to Eberron, are partially mechanical beings. For age, the rules say:

The maximum lifespan of the warforged remains a mystery; so far, warforged have shown no signs of deterioration due to age.

So a DM would be perfectly within reason to give one a lifespan of several thousand years or even more.

Magic items

The Ring of Winter listed in Tomb of Annihilation will prevent aging.

You, as DM can also feel free to invent any kind of item or artifact that can prolong life as well. Just be sure you are fine with having it fall into the PC's hands or have some sort of plan to have them need to destroy it.

In fact...

Make something up

If none of the RAW options above satisfy you, make something up. You are the DM and you have the power to change your world to contain whatever effects you want. Perhaps, the villain found or made a powerful artifact. Or maybe they found a sacred site or ritual that allows them to be long-lived. The options are really endless and really, there shouldn't be much of a concern here because the long-lived villain is a common trope and is not likely to be a problem for players to accept even without explanation.

Or even better, you can mix and match from all these options to create the long-lived villain of your dreams.

Use a monster.

There are plenty of creatures available that live for very long times. I won't bother listing all of them, but to name a few:

Dragons live a notoriously long time and can transform into humanoid form.

Fiends (read: Devils) are generally ageless*1

I'm unaware of an age limit for Hags.

Alternatively, MacGuffin

If you're the DM and you want the BBEG to have lived for thousands of years, then he's lived for thousands of years. Literally just make up a reason (try to make it a good reason); demonic pact, other worldly blessing, unicorn blood, whatever.

If you're determined for a RAW humanoid... a handful of class mechanics

As another user mentioned, cloning can keep you young for relatively little cost.

Having a druid cast Reincarnate on you after you die. This is risky because you have to trust someone to actually cast the spell, which is harder for a BBEG to achieve.

Being an 18th level druid slows aging by 90% (1 year for every 10). Combine this with an Elf or Dwarf and you can easily live thousands of years.

Honorable mention: being the target of Imprisonment. You don't age while imprisoned.

*1 The Monster Manual calls them immortal

The Epic Boon of Immortality

The DMG (on page 231) describes an optional system of further advancement for characters at 20th level called epic boons:

An epic boon is a special power available only to 20th level characters. Characters at that level gain such boons only if you want them to and only when you feel it's appropriate.

[...]

Epic boons can also be used as a form of advancement, a way to provide greater power to characters who have no more levels to gain. With this approach, consider awarding one epic boon to each character for every 30,000 XP he or she earns above 355,000 XP.

One of the boons described on page 232 is the Boon of Immortality:

You stop aging. You are immune to any effect that would age you, and you can't die from old age.

These options are intended for player characters, but there is no reason you cannot also use them for your NPCs. This method does, however, presume your NPC is a 20th level character, so your players would have to be quite high level to stand a chance against them.