Which is the most customizable BJD you know of?

Well there's the Iplehouse full choice system which is open from the 1st to the 15th of every month and allows you switch and customize their SD sized dolls. if the neck size fits, they'll allow you to make pretty much anyone on any body. I think they offer fantasy parts too and offer even some "limited" doll sculpts every month.
The Popovy Sisters also have a wide range of bodies available for their dolls to choose from including different jointing though all their dolls are female.
Fairyland also has their a la carte system which allows you to choose different bodies and headsculpts for their minifee line. You can pretty much mix match anything in that line and if you get the fairyline you might be able to find a second hand fantasy part to add to the body. though they don't have the same choices in resin color
Then there's companies like Angell Studio and Doll Family A which can allow for custom resin colors for an extra price if you ask

and I almost forgot about bobobie and resinsoul! They'll pretty much let you mix match anything including custom resin colors if you ask

 

Resinsoul by far but you have to email your request as opposed to choosing from menus. Kittyblue has some great examples on her YouTube channel of super custom dolls she’s gotten from them. They're open to doing all kinds of things: custom colors, changing joints, extra limbs/eyes, custom bodies, custom faceups, etc.

 

Volks does a very nice full choice system, from which one can choose head, body, arms, legs, feet and hands, eyes, a wide range of faceups, and their Satogaeri Services as an added bonus if one is interested,

 

Twigling has a nicely modular body configuration that gives you 4 different heights (2 torso lengths/2 thigh lengths) between 60-67 cm. Then also 4 busts, 3 hands, and 6 standard resin colours, but can get other colours their caster offers for a fee.

 

The new 1/3 girls from Doll Family H is pretty costumisable. There is a 58cm and a 63cm type body with the same cool options. You can choose from 3 breast sizes, and choose from two or three part torso, and with or without thight joint. I really like that. Oh and if you want her to wear high heels there are option for heel feet or heel leg. I think it's really nice from DFH that they allow such options. Especially the joints. If I want another girl doll ever I absolutly sold on the 64cm type girl body.

 

twigling's whole "pick torso/bust/legs" mixing and matching is pretty dang revolutionary I think.
Wish more companies could do that.

Resinsoul will mix and match anything you want but that's not a standard option, you have to email and ask so doesn't really fit the criteria of the question I don't think. But they will add eye sockets, close eyes, mix ears (you can have one sculpt's ears on another sculpt's head), limbs (you can have double jointed arms on the single jointed torso or one of their MANY different hands on any body), colours and so on. They'll also add limbs and have at least once modded a doll to have two necks so it could have two heads. Which is awesome. But yeah, they're not standard options on the website.

I LOVE modular body ideas.

A fair few do al a carte things where you can put any head on a selection of bodies but that's about as far as it goes. Body a or body b you know? Luts have about 5 bodies for each sex for their kid delfs though, but i'm not certain how varied they are in shape. Likewise Blue Fairy have 4 or 5 different options (I think?) for their tiny fairies which represent child, teen, older teen and more stylised. however they don't let you put any head on as they have their monthly selection so each month you'd only be able to pick from a half dozen heads. But you can put them on ANY of the bodies. Likewise they have a couple of bodies for their yosd which are different heights.

I can't think of that many that aren't one person artist cast ones that bother to do torso, limbs, hands, feet etc separately though.