I have killed at least one elk with each of the following rifles:

.243 Winchester
.25-06 Remington
.270 Winchester
7mm Rem Mag
.300 Savage
.308 Winchester
.30-06 Springfield
.300 Win Mag
8mm Rem Mag
338 Lapua Mag

Some were with handloads, others were with factory ammo and at various distances. The bulk of these were taken with my .30-06 Springfield.

I have personally witnessed others kill elk with all of the above cartridges and each of the following rifles:

6mm Remington
.250-3000 Savage
6.5 Creedmoor
6.5x284 Norma
6.5 PRC
.264 Win Mag
7mm-08 Remington
7x57mm Mauser
.30-40 Krag
.300 WSM
.300 RUM
.338 Win Mag
.375 H&H Mag
.444 Marlin
.450 Marlin
.45-70 Government

My personal preference is for at least a .270 Winchester minimum, but as I just said, cartridges as small as the .243 will do it given the bullet going into vitals.