totally agree with shevek...
Well, not unless there's a typo here:
EOR cannot be combined with LOR, since the cost of developing two launch vehicles will be prohibitive, if we’re talking about going into space in a big way.
If there's no typo there then I don't quite follow. I thought my point above, which may well be in error but is what I perhaps foolishly meant to say, was that the two rendezvous modes
can be combined. From what e of pi tells us, if we insist on using EOR to enable a direct lunar landing and ascent, we're either going to need Saturn V's, in quantity, or legions of smaller vehicles, to assemble a much more massive vehicle for transLunar injection. Since we are piecing together a ship from multiple payloads to orbit anyway,
why not make it a LOR modular mission? It's a hassle and a risk to be Legoing together a spaceship with no space station to serve as a platform, but I'd think that if a LOR ship is that much lighter there'd be fewer steps involved than assembling a more massive one, even if much of what we are snapping on is just clusters of fuel tanks.
Let's see, the OTL LOR system was 30 tons for the CM/SM, 14 for the LEM, and the Saturn V third stage (=Saturn 1B 2nd stage, called the S-IVB) was 120 tons on launch. Breaking all this down into 15 ton units, we have 2 plus 1 plus 8

! Clearly the S-IV is the sticking point here; trying to break its integrated fuel tankage into clusters that fit into 15 ton loads would introduce a fair amount of parasitic mass, not to mention complexities that are extra vulnerable possible points of failure! And who exactly would be methodically piecing together all this?
Actually of course the S-IVB would not mass 120 tons for transLunar injection; much of its propellent would be used up in the course of the third launch stage to put the assembly into parking orbit. Piecing things together from
two separate Wiki pages

the third stage's share of the orbital burn was 150 seconds while the translunar injection was 350 seconds, so actually a translunar booster that only has that job, having been launched to parking orbit as cargo, would presumably have to have 7/10 the fuel. The OTL stage had a bit over 105 tons of fuel, so say the Artemis injection rocket now needs 71 tons. Call its all-up mass after achieving orbit 90 then.
So, 135 tons all up, plus any extra that comes from splitting OTL integral units into modules. Well, let's make it a round 140 tons, and think in 20 ton units--we can presumably slim down the Service Module by hauling 10 tons of its 18 tons of fuel for the main engine on one of the other launches and then snapping it on during Earth-orbiting vehicle assembly. 5 of those tons might go up with the LM load. That leaves 5 more launches out of 7 to assemble the translunar injection rocket. Perhaps we can offset some of the inefficiency of the modular assembly of numerous tanks by successively blowing them off as they are exhausted.
Or, we can think in 5 30 ton units, or 4 34 tons, or 3 44 tons, or finally two 68 ton units. The bigger the units the fewer the launches and assembly steps and modular inefficiencies, but the bigger the standard launchers have to be.
Even with two launchers, each half as capable as the Saturn V, we are quite awkwardly subdividing the translunar rocket, as its fuel alone would mass more than 68 tons; the 45 tons of the spacecraft won't fill out the other load so we are carrying along fuel in a penny packet to tack on to the main injection rocket. Well, this can be done most naturally as a second stage.
Still a 68--round up to 70!-- ton to parking orbit capability is embarrassingly high for any more routine use. Designing a half-Saturn V might beg the question of why not design a whole one and be done with it! The closest thing to a natural division of the load to my eye is to have three launches, each sending up 50 tons (kicking it up to allow for modular inefficiencies and then hopefully if we have any margin left over, Artemis might be a bit bigger a spaceship in terms of useful payload) to assemble a 150 ton all up Lunar vehicle. That's 50 for the manned modules, and two loads to snap together possibly by remote control from the manned unit standing off to the side to make the translunar rocket. Which the manned units could again most easily join "upside down"!
Or maybe, snap two parallel rockets each with integral fuel tanks and their own engines together side by side, and hook the CM onto the "bottom", with the two injection engines angled out a bit so the blast doesn't hit the manned ship, and let the manned elements hang from the two rockets between the engines? Or something like the Shuttle arrangement, with the manned units snugged onto one side of the parallel joint between the two parallel units riding sidesaddle?
Something to consider here is that since the S-IVB stage used two cryogenic propellents, liquid hydrogen and oxygen, and since LH needs to be kept especially cold, there is a time limit as the propellants, the hydrogen especially, will be boiling off in the sunlight while someone maneuvers them into linking up. Aside from the arguments involving extra parasitic mass associated with breaking the translunar engine into even smaller segments, presumably the more pieces there are the more time it will take to assemble them and hence the more waste of propellent, also the more steps there are the more uncertain the timetable is to complete the assembly process. I think maybe three steps, first launching the manned unit then two parallel (or serial, snapped one on top of the other with the manned unit then backing on to a link up with this stack) injection rockets at the same time or in very quick succession might not be a lot more time consuming than the 2 orbits the standard Apollo mission profile provided for OTL. Perhaps, if we can do multiple launches of many smaller rockets simultaneously, we can break it down into 6 25 ton units--first the CM/shrunk SM, then the LM to be scooped up on the nose as OTL (but braced there, because it is going to be under thrust, and a fuel line for its share of the SM fuel has to be hooked up too), then 4 simultaneous launches of 4 translunar rockets, to be hooked onto the sides of the SM and braced to form a collar of rockets--I think that's the lower limit of reasonable masses to be pieced together. Subdividing them further suggests breaking up the main mission modules themselves and I think that's going too far!
Alternatively instead of LH-LOX engines, the translunar rockets can be fueled with the less efficient but non-cryogenic (but toxic!) fuels used in the Titan rockets and others, these have also the virtues of being denser hence needing less volume, and being storable we buy more time for a more leisurely assembly of the craft. But being less efficient--well, it's getting late and I'm getting confused, but we might well need a fifth and perhaps even sixth injection module, another 25-50 tons. But what's two more launches?
The Wiki article on the Apollo program says as many as 15 launches were contemplated in some EOR schemes; the stub article on EOR says the rockets involved were to be "half" the size of the Saturn V (!) which would imply no more than three launches I'd think.
Even using a less efficient (but much more compact and less fragile, a major consideration for this laborious assembly project) storable fuel mix, I don't think we'd need more than 8 launches. But if we have to launch each one one at a time, that's a lot of orbits for the Artemis crew to be making as they come up--half a day or more. If the plan is to launch up to say six translunar thrust units before the two manned units go up, use ground control to remote-control the six into say two clusters of three each for the manned craft to assemble by attaching first one than the other to the SM sides and then using some kind of actuator link them with braces, after first spearing the LM, stashing it on the nose and bracing it there for these maneuvers--well that might save some orbital time and get the task down to something we'd wish on three men in a working day.
This scheme of mine then relies on having 25-ton to orbit boosters, which strikes me as a reasonable capability for future orbital missions and just right for a lightened Apollo-type CM/SM, suitable for orbital missions. Or a 25 ton spaceplane of some kind, or 25 ton cargo packets.
Having a space station of some kind, even a very small one, to serve as the construction platform for this EOR assembly, strikes me as another good idea. Say, one pieced together from two or three specialized 25 ton modules, manned by an Artemis CM mission, that successively remote-controls in each launched translunar module, slides it down tracks to the assembly of the prior ones forming a ring, then when the ring is all assembled, slides it back to the tip of the station. Meanwhile the second to last launch sends up the LM, which the station crew remote-controls in to stick on the other end of the station. Now the CM/SM stage of the moon mission comes up last, backs up to the station tip with the engine ring, gets secured to a mounting there, the station slides the ring to its attachment position on the SM, releases the assembly, now both station and spacecraft rotate around so the LM is ready to link onto the nose of the CM. The station crew helps make sure the linkages and braces are all deployed and secured correctly, then the station and spaceship maneuver away from each other, and when well separated the translunar rockets do their thing!
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So then, we'd need: one each space station requiring 2 launches to send up;
One station assembly mission, involving a lightened CSM unit for orbital work to go up to supervise linking the station modules together, man it to check it out, and then perhaps immediately prepare for an Artemis mission to be sent up shortly if all goes well there.
3 launches so far. Maybe 4 if the task of facilitating the moon shot is too much after putting the assembly station together.
Six successive launches of translunar propulsion units, assuming storable fuels; each one is remote-controlled to dock with the station and get assembled. Total 9.
One LEM launch, 10.
The CSM launch, 11.
Once the Lunar mission is on its way the station team can come down, as there is little they can do to help the moon mission if something goes wrong.
Subsequent missions will require 9 each launches, 8 for the moon mission and the first one for the station crew.
Thus we need to launch a total of 225 tons to orbit for each moon shot, as opposed to OTL 135, but that's mainly because of using storable fuel and needing a human crew on the assembly jig station. If we don't need an assembly station, or if it can be remote-controlled, we can eliminate one of those 25 ton units; if we can slim down the translunar rocket assembly by using liquid hydrogen because we can speed up the assembly process, we can eliminate 2 more, and that would bring us down to just 15 more tons than OTL, which seems reasonable to me.
The standard booster, instead of having to raise 135 tons to orbit, only has to lift 25 and so should scale down from 3000 tons to a bit over 555 tons on the pad, call it 560 or even 600 if we suffer from some inefficiencies or are locked in to a lower state of the art by freezing the design some years earlier than Saturn V was OTL. By that same token, these rockets ought to be ready for testing and then use that much earlier.