Qual rei tomou posse da Pedra de Orthanc?

Aragorn mantém a posse da pedra. A seguinte citação de Many Partings dada por @MattGutting é suficiente para estabelecer isso.

'Only one now remains that you could use,' answered Aragorn, 'for you would not wish to see what the Stone of Minas Tirith would show you. But the Palantír of Orthanc the King will keep, to see what is passing in his realm, and what his servants are doing. For do not forget, Peregrin Took, that you are a knight of Gondor, and I do not release you from your service. You are going now on leave, but I may recall you. And remember, dear friends of the Shire, that my realm lies also in the North, and I shall come there one day.'

where he [Aragorn] proposed to set up again the palantir recovered from Saruman.

Por que aqui? Por que não confiar a pedra de Orthanc ao seu mais próximo e poderoso aliado, Éomer, e colocá-lo em Edoras? Por que não colocá-lo em Annuminas, para ficar em contato com seu reino do Norte, ou mesmo colocá-lo em Dale, já que futuras ameaças a Gondor provavelmente viriam do Oriente? Isso levou algum trabalho, mas a resposta é dada na nota 18 do The Palantiri (Contos Inacabados novamente).

The greater palantiri could look much further than the lesser; for the lesser the 'proper distance' was of the order of five hundred miles, as between the Orthanc-stone and that of Anor.

(Ênfase minha). Somente devolvendo a pedra Orthanc ao seu lugar original e designando um diretor adequado poderia Aragorn estabelecer um meio de descobrir imediatamente o que estava acontecendo em Arnor. As principais pedras (Amon Sul e Osgiliath) foram perdidas. Uma única pedra menor, mantida em Minas Tirith, teria alcance insuficiente.

Não há evidências diretas . Orthanc, Isengard e a Pedra eram, naturalmente, propriedade de Gondor e, portanto, do rei. Nesse mesmo capítulo que você cita, Aragorn faz uma grande dose de especificamente cedendo Orthanc e Isengard aos Ents:

[Gandalf:] 'But the Tower of Orthanc now goes back to the King, to whom it belongs. Though maybe he will not need it.'

'That will be seen later,' said Aragorn. 'But I will give to Ents all this valley to do with as they will, so long as they keep a watch upon Orthanc and see that none enter it without my leave.'

Ele não diz nada específico sobre dar o palantír a ninguém - em particular, ele certamente parece não ter dado aos Ents (que provavelmente não o teriam desejado em nenhum caso); e parece improvável que ele tivesse acabado de deixá-lo em um prédio trancado. O contexto da citação que você dá na sua pergunta, no entanto, pode dar uma pista:

'Only one now remains that you could use,' answered Aragorn, 'for you would not wish to see what the Stone of Minas Tirith would show you. But the Palantír of Orthanc the King will keep, to see what is passing in his realm, and what his servants are doing. For do not forget, Peregrin Took, that you are a knight of Gondor, and I do not release you from your service. You are going now on leave, but I may recall you. And remember, dear friends of the Shire, that my realm lies also in the North, and I shall come there one day.'

( Retorno do Rei , Livro VI, Capítulo 6, "Muitas Partituras"; ênfase adicionada)

Isso não está claro.

It must however be noted with regard to the narrative of The Lord of the Rings that over and above such deputed authority, even hereditary, any "heir of Elendil" (that is, a recognized descendant occupying a throne or lordship in the Númenórean realms by virtue of this descent) had the right to use any of the palantíri. Aragorn thus claimed the right to take the Orthanc-stone into his possession, since it was now, for the time being, without owner or warden; and also because he was de jure the rightful King of both Gondor and Arnor, and could, if he willed, for just cause withdraw all previous grants to himself.

^{_{Unfinished Tales Part 4 Chapter III: The Palantíri}}

No entanto, não está claro se isso se refere a Aragorn reivindicando o Orthanc-stone em The Two Towers , ao invés da linha em discussão:

'There is one who may claim it by right. For this assuredly is the palantír of Orthanc from the treasury of Elendil, set here by the Kings of Gondor. Now my hour draws near. I will take it.'

Gandalf looked at Aragorn, and then, to the surprise of the others, he lifted the covered Stone, and bowed as he presented it.

'Receive it, lord!' he said: 'in earnest of other things that shall be given back. But if I may counsel you in the use of your own, do not use it - yet! Be wary!'

^{_{The Two Towers Book III Chapter XI: "The Palantír"}}

Infelizmente, não parece que Tolkien tenha resolvido a ambigüidade (ou, na verdade, se ele alguma vez a considerou ambígua); não há nada que eu possa encontrar em Cartas ou História da Terra Média que surja de uma forma ou de outra.

É inquestionavelmente Aragorn quem manteve o Palantir, não apenas pelas razões já descritas, mas também porque Gondor manteve a torre como um exclave

the Stewards retained under their own rule the Tower of Orthanc and the Ring of Isengard (Angrenost); the keys of Orthanc were taken to Minas Tirith, the Tower was shut, and the Ring of Isengard remained manned only by a hereditary Gondorian chieftan and his small people.

-- Appendix II, The Battle of the Fords of Isen, Unfinished Tales