This is not another "Who was Tom Bombadil" question. Close to the end of ROTK Gandalf tells one of the Hobbits that it's time for him to go and have a long talk with Tom Bombadil. Was there anything significant to this talk? Does anyone know what it would be about, etc?
Perhaps Gandalf wanted to investigate any being who could not be affected by the One Ring, as it contained the power from a Maiar (Sauron). Remember how Bombadil remained visible when he placed the Ring on his finger, and how, in fact, he had power over IT, causing it to disappear and reappear.
With the understanding that
- Bombadil was clearly depicted as the oldest living being in Middle Earth (He tells the Hobbits this in their encounter with him ... "Tom was here before the river and the trees"(FoTR))
- Gandalf was one of the Maiar, which were spirits created before Middle Earth took the shape it was in when the elves first awoke (Olórin he was called before Gandalf, cited in LOTR and The Silmarillion)
it is easily argued that these two were kindred spirits from waaaaay back.
Tolkien humanizes Gandalf in his role as "old man", in that he hungers to have the freedom (after his long labors against Sauron, also once one of the Maiar*) to sit down over a pipe and have a long discussion or reminiscence with someone more like him, someone who remembers the old days.
There are very few beings on Middle Earth who could fulfill that hunger: Bombadil was one such.
*As @corsiKa pointed out, Gandalf had recently defeated three other Maiar: (Durin's Bane aka the Balrog (corrupted ages ago by Morgoth), Sauron, and Saruman (who had been corrupted by the desire for Power, or as Gandalf put it in LOTR, wanting to "become a Power")). These three were, back during the Great Music described in The Silmarillion, originally spirits of the same kind. Gandalf and Tom Bombadil might very well be "the last of the breed" in Gandalf's reckoning, and one of the few with whom he could share the old stories.
** FWIW, his Sindarin name Iarwain Ben-adar (Eldest and Fatherless) is another point to him being the first being on Middle Earth as we know it.