Criteria:
any class combination is ok
feats are allowed
magic items are ok
Constraints:
no use of the wish spell
no homebrew
can have up to 5 people assisting
turning yourself into an elemental doesn't count
How many elementals can you control at once?
You can have control of as many elementals as you like by using the true polymorph spell which says:
Choose one creature or nonmagical object that you can see within range. You transform the creature into a different creature [...] whose challenge rating is equal to or less than the target's
Then all you need is to get permanent control of any kind of creature, for which finger of death is your friend:
A humanoid killed by this spell rises at the start of your next turn as a zombie that is permanently under your command, following your verbal orders to the best of its ability.
So, all you need to do is kill someone with finger of death and then true polymorph the resulting zombie into a CR 1/4 elemental such as a mud mephit.
While a swarm of mephits is terrifying in its own way, any summoner worth their salt wants to command more powerful elementals than that. Depending on how your GM rules on the interaction between magic jar and true polymorph, you can accomplish this by following these steps:
In order to follow the rest of the procedure, this is needed since otherwise zombies aren't able to cast magic jar (possibly because they have no soul, possibly because they have no living body; see this Q&A for details)
Cast true polymorph onto a more powerful elemental of choice to make it a humanoid.
Have the former zombie use a scroll of magic jar against the humanoid to possess it (a spell gem is a more renewable option if scribing spell scrolls is prohibitively expensive)
Repeat step 2 until the magic jar spell is successful, and the former zombie possesses the transformed elemental.
Stop concentrating on the true polymorph spell that affects the elemental. Once magic jar is in effect, the possessed target no longer has to be a humanoid since every reference in the spell thereafter only says "possessed creature". The target need only be a humanoid for the initial possession.
I'd like to reiterate that this ruling on how magic jar and true polymorph interact is debatable, so ask your GM before attempting this kind of method
Our character:
1 level Cleric to qualify for Book of Exhalted Deeds we are attuned to and have read.
19 levels in Divine Soul Sorcerer with the Twinned Spell metamagic. This also lets us learn planar binding as a sorcerer/cleric spell. †
Boon of High magic for an additional 9th level spell slot
We're an elf so that we can trance for 4h instead of taking an 8h long rest
We have helpers who can summon elementals. Which doesn't matter, we just need 2 at any one time.
9 pearls of power and a mountain of gems each worth 1000 gp.
The key is twinning an upcast planar binding. When upcast it binds the two targeted elementals to our plane and importantly command for X days. If we then cast planar binding on the previous X days we command 2×X elementals for each spell level we can do that.
This is where our Boon and Book comes in. High Magic gives us an additional 9th level spell slot and Book of Exhalted Deeds make all our slots count as one higher. The only effect of our 9th level slots giving 10th level slots is it costing us more to twin them.
Using our 10 9th to 5th level spells‡ we need 77 sorcery points. We get 19 from our sorcerer levels, 31 from converting our 1st to 4th level slots, and 27 from converting the slots regained using the pearls of power.
This results in the following twinned planar binding castings per day: 3×9th, 2×8th, 2×7th, 3×6th. This lets us accumulate a number of elementals up to $$ 3\times366\times2+2\times180\times2+2\times30\times2+3\times10\times2 = 3096 $$
At least for the last days, we have spendt 10h casting planer binding, 8 hours attuning to new pearls and 4 hours in trance.
†: There's an argument here that you'd need to prepare planar binding as a cleric for it qualify as a cleric spell for Book of Exalted Deeds. If so, we need 9 cleric levels giving us 8 fewer sorcerer points, but grabbing another pearl of power, we can still have enough points for twinning all but the last 6th level casting. This means the ruling nets us only 10 fewer elementals in the end.
‡: I have assumed you can't cast a 5th level spell using 4th level slots, even with BoED. If you (and your DM) think you can then it would add a measly 3 elementals to the sum.
There's an iteration of this method which uses Wild Magic Surges to regain spell slots (the 59-60 result) and sorcery points (99-00 result). For it we are at least 9 levels of cleric (for planar binding) and 9 levels of Wild Magic Sorcerer (for enough sorcery points), the last two is some full caster levels. We have Boon of High Magic and Boon of Spell Mastery. For the latter boon we choose a spell which requires an attack. For that attack roll we use Tides of Chaos, and on subsequent castings our DM has us roll a Wild Magic Surge to regain the use of that feature. (There are other ways to obtain this loop).
The outcome is that we repeat this until we've hit 59-60 and 99-00 (on average 50 tries or 5 minutes). So for each of these we can cast a twinned 9th level planar binding. Our day is then 4h of trance, 2 hours of WMS loop (we also have some grace time in here to ride out the negative effects our helpers can't dispel for us. Some pink-bubble mouth to be expected) for 18h of casting planar binding letting us accumulate $$18\times366\times2=13\,176$$
Elementals. Though do note that your DM needs to approve each time you roll WMS, though that seems a little void in the circumstances.
David Coffron gets a massive thank you for helping me hash out some of the real cheese of this answer.
Unless magic items let you summon/conjure elementals differently that the spell Conjure Minor Elementals page 226 of PHB, concentration is required. So, the most you could control, if you cast the spell at its default level, is 8 of challenge rating 1/4. Upcasting at levels 6 and 8 allows double or triple that number. (Thank you someone_evil.) Conjure Elemental page 225 of PHB summons a single elemental servant.
