In the past I have always traveled with one or two other hunters and we were always armed with overlapping calibers. This year due to circumstances beyond anything I ever anticipated I will hunt alone. My double rifle will not be done in time to go therefore I will take the old reliable pair my 404 Jeffery and my 375 H&H. We always had a 300 H&H along for the smaller stuff that was included in my wifes kit. Should I also bring this third rifle. I will be in the Rungua for 21 days for a full bag safari. Will I miss the smaller caliber shooting the 375/404 combination and is it worth the hassle of the third rifle and another gun case to worry with?

...twenty-one trips/hunts to Africa, and I've NEVER had an occasion to utilize more than two different rifle/calibers. 99% of the time, one will suffice for everything. If you're happy with your .375, and .404 I see no reason to burden yourself with a third rifle.
FWIW...for African hunting, I've downsized my battery to one rifle; a .416 Taylor. Kills the little stuff, as well as DG.
Good luck

I was in RSA last year and used only my 375 H&H (Ruger Deluxe Mag with 1.75 x 5 Burris)

Buff - 28 yards, 300 gr Swifts & TBC Solids. Swift hit both lungs and heart, solids slowed him; I think a 400 caliber would have put him down on the first shot, the 375 swift nearly did. I would have used the Norma 350 gr PH ammo if available then.

Plainsgame -
300 gr Swifts
Blue Wild 150 yards
Nyala 110 yards
260 grain Accubonds
Red Lechwe 350 yards
Common Reedbuck 160 yards
Cape Kudu 150 yards

Caracal - 300 grain Solid (10 yards)

I think your .404 and .375 is a great combination just in case you run into something big when you are hunting with your little gun. If you are going to do a lot more plains game then a .404 and .338 would be what would fit my needs, or even the .404 and a .300 mag of some sort.

good luck on your hunt,
namibiahunter

is it worth the hassle of the third rifle and another gun case to worry with?

NO.

I bring two rifles.

On DG hunt I bring my scoped stainless M-70 in .375 and an iron sighted heavy either a .458Lott or my .470NE double.

On a plains game hunt I bring my .375 and a lighter rifle like a .30-06.

I would take every rifle I own with me if that were possible. But with you traveling alone an extra case is a big worry. So, just take two; the .375 H&H and the .300 H&H.