Well, I don't actually want to go through the entire process to figure out just how huge you could make your Frodo, but I'll go ahead and list some of the cards involved. (Also, I'm assuming all cards are fair game, as you didn't specify a format.)
Oh, and... this wouldn't be a permanent strength boost for my companion, just a temporary thing. So, maybe this doesn't count at all and you should just stop reading now. =P
For my Ring-bearer, I'd go with
Frodo, Weary from the Journey with
the Ring of Doom. I'm assuming he's skirmishing a non-Hunter minion of strength 8 or less. =P
His partner in crime will be
Merry, Friend to Sam.
Merry will have
Escape,
Frodo Gamgee, and
Little Golden Flower for +vitality. There will be 4
Everyone Knows in play. We'll also have
Southfarthing Leaf and every bit of pipeweed you can imagine.
And, of course,
There and Back Again and 4
Long Live the Halflings/Don't Let Go.
The big thing is the 4 Shire Countrysides and the
Frodo's Cloak. This allows you to heal Merry 4 times, getting two extra sets of exertions out of him.
Basically, use a bunch of crap to boost Merry as high as you can (heck, even toss in some
Goldberry exertions!), then exert him to add his strength to Frodo multiple times. You should be able to add his strength 8 times just with the cards I listed, and I imagine there are even more permanents that will allow you to heal him or remove a burden during the skirmish phase. Oh, and don't forget 4 copies of
Sudden Fury, assuming they call came into play at site 1 and there were ridiculous amounts of skirmishes being fought at each site (even "extra" fierce skirmishes, like with
Return to its Master and/or
The Witch-king's Beast). Use all of this stuff to pump Merry, then multiply that by 8 and add it to Frodo. =P
But, once again, that's not a PERMANENTLY large companion, like Durin in the opening example.