A Devotion Paladin 10/ Moon Druid 10 character (level 20 total) can get 10+ "permanent" immunities, and a few more that are conditional (potentially 13 total)
Before we go any further, I wanted to specify why "permanent" is in quotes in the title of this answer. There are ways to end these immunities in most of the cases, but my criteria for "permanent" is that it can be maintained perpetually by the character in question for the extent of their natural lifespan, only using resources which can be recharged at as fast or faster a rate than they are used (like uses of Wild Shape) or that are easily restored (like lantern oil).
With that out of the way, let's get down to specifics.
8 "Permanent" immunities from Level 10 Circle of the Moon Druid
Druids don't get any conditional immunities inherently. But a level 10 Circle of the Moon Druid has the Elemental Wild Shape ability, which states (PHB, p. 69):
Elemental Wild Shape At 10th level, you can expend two uses of Wild Shape at the same time to transform into an air elemental, an earth elemental, a fire elemental, or a water elemental.
An Air Elemental (MM, p. 124) has the following features:
Condition Immunities exhaustion, grappled, paralyzed, petrified, poisoned, prone, restrained, unconscious
That's eight condition immunities, right off the bat.
A druid can stay in this wild shaped form for half their druid level, meaning 5 hours. So long as you take a short rest starting in less than 4 hours from when you transformed, you'll regain your uses of Wild Shape, and can maintain the form for another 5 hours.
Now, you might have noticed that the elemental is immune to the Unconscious condition, meaning they cannot sleep, which means they are incapable (under the rules) of taking a long rest by default. However, since they are also immune to Exhaustion, they will not suffer exhaustion from missing long rests, and thus can maintain this form, and the resulting conditional immunities, indefinitely.
2 "permanent" immunities from level 10 Oath of Devotion Paladin
A level 10 Oath of Devotion Paladin has the following features (PHB, p. 86 and 85 respectively):
Aura of Devotion Starting at 7th level, you and friendly creatures within 10 feet of you can't be charmed while you are conscious.
Aura of Courage Starting at 10th level, you and friendly creatures within 10 feet of you can't be frightened while you are conscious.
That's two more perpetual condition immunities that the Air Elemental didn't have already (since wild shaped Druids retain the benefits of their classes). And since the Air Elemental is immune to the unconscious condition, these auras will always be active.
The Paladin is also immune to disease (at 3rd level), which isn't a condition, but is worth noting.
Condition immunities that are... conditional (sorry, can't think of a better phrase for that)
Tasha's Cauldron of Everything introduces a new fighting style for Paladins (page 52):
Blind Fighting You have blindsight with a range of 10 feet. Within that range, you can effectively see anything that isn't behind total cover, even if you're blinded or in darkness. Moreover, you can see an invisible creature within that range, unless the creature successfully hides from you
Although this benefit is perpetual, it is also conditional for a few reasons. First, it has a very limited range (10 feet). It also doesn't technically make you immune to the blinded condition, but it does mean you won't suffer most of its effects. It's also conditional on a DM's interpretation of the rules, because Wild Shaped characters don't usually retain any "special senses" unless their new form has them. However, since your original form's blindsight is not necessarily biological, but a result of training, your DM may allow it to work while Wild Shaped.
The character could also have a Ear Horn of Hearing, which suppresses the effects of the deafened condition on you" (XGtE, p. 137), thus giving you immunity to the deafened condition. However, it does so when "held up to your ear" (ibid). Does an air elemental have ears? That's hard to say. A DM will need to make a call. So this immunity is also conditional.
Finally, you could invest in a Lantern of Revealing, which states (DMG, p. 179, bold added):
While lit, this hooded lantern burns for 6 hours on 1 pint of oil, shedding bright light in a 30-foot radius and dim light for an additional 30 feet. Invisible creatures and objects are visible as long as they are in the lantern's bright light.
Thus, this character could also functionally attain immunity to the Invisible condition (or at least not gain benefits of it), provided they kept the lantern lit and on/near their person (possibly needing to carry it as an air elemental). However, they would need to continue to provide the lantern with oil, and it's possible that refilling this lantern would require you to first extinguish it (depending on how it is constructed). So a DM could rule that there is a brief period during which your character can become invisible, making this immunity also conditional.
It's worth noting that even the "permanent" condition immunities mentioned earlier can be ended by the right set of circumstances. The easiest way to end the wild shape (air elemental) immunities would be dealing enough damage to force the druid out of their elemental form, or subjecting it to an antimagic field. Having done that, you could deal enough damage to the character that they fall unconscious, which would also end the Paladin's immunity to being charmed or frightened. It's also worth noting that this build requires the use of two magical items (providing one conditional immunity each), and each item may need to be held in a hand (or its rough equivalent) by the air elemental, meaning its options for further engaging with its environment are quite limited.
Still, we have a build here that can give indefinitely sustainable immunity to between 10 and 13 conditions. Since there are only 15 conditions listed, that's a pretty good place to be.