Of course that depends mainly on which country and which language is meant. Some names carry on the ancient names of countries and/or tribes, other ethnonyms convey the history of a long-period quarrels (settled by time or not).
That is to say, the Japanese and/or Chinese names of countries are fabulous, but there are also some more examples from IE languages in Europe.
To start with, there is a language with no names for lands like Checkia, Hungary, Germany and Italy, and that language is Polish. The literal meaning of the country names would be the Checks (Czechy), the Hungarians (Węgry), the Speechless ones (Niemcy) and the Italians (or Włochy, as named by the Speechless ones, and please don't ask me how it comes to be) respectively. The names of Lithuania and Latvia in Polish also mean 'a group or a tribe' with collective sufixes: Litwa and Łotwa.
Speaking on which, and moving along the European map, we meet some more examples of exotic names. Lithuanian, this ancient language of Baltic states, offers such a collection of ancient onomastic survirvors as Lenkija (Poland), Suomija (Finland), Vokietija (Germany; not a big surprise), and Latvian offers the hidden treasures of ages passed like Igaunija (Estonia), Somija (Finland again), Vācija (Germans seem to lose their perfect Clear toponym wherever they get into), the almost irrecognisable name of Sweden masked as Zviedrija and - surprise again! - the powerful Eastern neighbour Krievija a.k.a. Russia (although I used to think, up to this post, that Lithuanian shares this exotic variety of view on its occidental neighbour with Lithuanian).
The Swedish has nothing to surprise a linguist with, perhaps with the exception of Ungern (Hungary).
Estonia itself is a good example of being both Eesti and Viro in Finnish, which also keeps the old traditions of near-neolythic naming of now-extinct tribe names in such words like Venäjä (Russia), Ruotsi (Sweden), Puola (Poland), and calqued Belorussia (Valko-Venäjä), Itävalta (Austria, a.k.a. Eastern State) and Alankomaat *(Nederlands, or Neath-Lands)*. I am not mentioning Tanska here, for its original mode of Danmark is quite recognisable.
The Estonian names often rhynme with Finnish patterns, e.g. Rootsi (Sweden), Venemaa (Venäjä), and even if the Netherlands are Holland, being occasionally Madalmaad (Lowlands), the Danmark is still Taani, and the Norway is Norra.
Of other Finno-Ugric languages I am familiar with, Komi mostly copies the Russian names, with the exception of Russia (Rochmu), and Hungarian claims Italy to be Olaszország, otherwise adding the ország *(-land)* part to the rest of most other country names: Spanyolország (Spain), Németország (Germany), Lengyelország (Poland) and masked mysterious Magyarország (Hungary), not mentioning Észtország (Estonia), Lettország *(Latvia)* or Oroszország (Russia) together with Írország (Ireland). They are strange, but mostly regular.
What is not so regular, is the Ghaelic orthography, and here are some interesting Irish examples; An Eilbheis (Swissland), An t-Seic (Check Republic), Innis Tìle (Iceland), An Eadailt (Italy), and almost any other country the name of which is made irrecognisible by a magic spell casted with Irish orthography: A' Bhealaruis (Belorussia), Moldàibhia (Moldova) or An Laitbhe (Latvia). The Ireland itself becomes Poblachd na h-Èireann, the English is Beurla and England is Sasainn, but I think there is nothing outstanding about these facts.
Trivia: the Irish name for Englishmen would make most of Russians laugh, since Sasanach in Irish is pronounced almost like 'wtf' in Russian.
The Basque language shows a certain modesty in exoticism as compared to this Indo-European feast of onomastics; Errumania, Errusia, Eslovakia and Eslovenia are just Romania, Russia, Slovakia and Slovenia; Txekia is Checkia and Herbehereak are Neath-Lands again, but why Serbia, Suedia, Suitxa (Swiss) and San Marino have no initial *E*s? This is just unfair.
PS Of course, I am not referring here to Erresuma Batua, since the United Kingdom is a strange pair of words when translated into any other language; nor speaking I about Montenegro, which is sometimes translated into other languages as Black+Mountain, and sometimes, not.