Best cartridge.... tough one to answer, especially with specifics,

Generally, for b-e-s-t, "the biggest you can shoot comfortably and accurately" (up to 338WinMag or so, then its a bit overkill but kinda fun [375H&H sure drops the hammer on them]).

For just plain s-u-f-f-c-i-e-n-t, other considerations may apply.

Also IMHO, a 30-06 minimum. Iffolks can't handle a 30-06 then it sure would be nice if they could just take a pass on the elk hunting, but that is just my world.
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BTW, choice of equipment makes a significant difference in the equation for answering the question.

For example:
We just got back from the Pawnee National Grasslands from a day of shooting....
We were playing with the newly acquired 300WinMag Sendero with a muzzlebrake (never hada muzzlebrakebefore except on the 50 BMG and its more of a blast deflector than a brake), a high power scope with external adjustable windage and elevation turrets with MOA read outs ....

That supposedly mean, nasty, unmanagable "magnum" had less kick thanmy 270 (maybe more like a 6mm Rem) and thats shooting prone with the recoil pad resting more on my collar bone than tucked into myshoulder meat....

This was the first time out with this set up, we boresighted it the day before at 100 yards, and then after some adjustments to get it zero'ed at 200 yards, we cranked in the MOA adjustments off of the drop/windage table and I was surprised:
--500 yards, one 3.5 inch group (3) all in the six inch bull
-- 1000 yards, one 14" group (3)none in the six inch bull (a bit trickier with the wind andthelonger flight time)

Even though it is a 300WinMag, this is decidedly a gentle and agreeable little rifleand the possibilities sure seem interesting....

Who knows, perhaps my partner and I willbe hunting with Walker Game ears this fall, so we can hear better and can use a braked rifle with impunity. I've always been a 200 yard man, but this is looking to be a fresh, new, fun chapter to an "old book" thathonestly has beengetting a bit boring the last few years.

Contrary to popular belief, blasting them as close as 30 yards while hearing their breathing and the clatter of their hooves and seeing their black eyes, their near-black tongues, and their"sticking straight up stubby little tails" doesn't necessarily create the thrill or satisfactionI get froma 200 yard (or longer) kayo shot.

Equipment, technology, technique.... it all can make a difference in the "answer" to the "what is the b-e-s-t" question.... one recommendation I'd have though.... would be "hit them hard" AND "hit 'em where it counts." Damn, 500 yards was easy....this little 300WinMag Senderosetupisone fun little rig.
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That said, we have found that BJ's advice of 30-06 for newbies is sound advice as newbies generally are not long on equipment, technology or technique when it comes to elk. Our camp has had good luck with the newbie/30-06 combo. [180 gr.Partitions]