Strictly, it is possible that the only language that does not do without "to" is English.
Even if we discard what are obvious quirks of the language, such as its use as an infinitive marker (as in, "I want to eat" - compare Spanish "Quiero comer" or French "Je veux manger"). As user6726 says, there are several different prepositional senses of the word; it can express many different kinds of relations between words, which the average user manages without a second thought, but may be totally misterious for someone whose first language is different, for there is little logic in the cluster of senses of "to".
For instance, this trivial English sentence,
I am going to London.
Could be translated in two different ways into Portuguese:
Eu vou a Londres.
Eu vou para Londres.
(The latter implies the speaker is moving to London, while the former implies a quick trip.)
So, which is the "to" of Portuguese? A or para?
Given the quite anarchic nature of the set of prepositional sences of "to" (and most other English propositions), and the fact that other languages have equally anarchic prepositional systems (why English distinguishes between "at", "in" and "on", where Portuguese does so well with only "em"? Why does English use "on" as a synonym of "about", like here, while in Portuguese this would be "sobre", which can be "about", but also "over"?), the chance there is a language with a word that can be univocally mapped into "to" is infinitesimal.
Evidently, as we move to other languages, with even less relation to English, the confusion can only grow.
So, in principle, all languages, except English, do without "to": they either use different prepositions corresponding to each of "to" senses, or they use other syntactic devices, as noted in other answers and comments: adverbs, verbal affixes, nominal affixes, word order, different verbs, etc.
ETA glosses for your examples, in Portuguese:
- To eat. (infinitive) - Comer.
- I want to eat. ([verb] to [verb]) - Quero comer (it is just an infinitive, too).
- I went to the store. ([verb] to [noun]) - Fui à loja ("à" is the contraction of preposition "a" with article "a") ("to" = "a").
- It went up to the ceiling. ([adverb] to [noun]) - Subiu no teto ("no" is the contraction of preposition "em" with article "o") ("to" = "em").
- Another way to put it is... ([noun] to [verb]) - Outra maneira de dizer é... ("to" = "de").
- ten to six ([noun] to [noun]). Dez para as seis ("to" = "para"). Alternatively, Seis menos dez (no "to"; literally, "six minus ten").
Aditionally, European Portuguese uses the prepositon "a", which expresses some of the relations "to" expresses in English, as a marker... not of infinitive, but of what could be called a "composite gerund": "Estou a comer", "I am eating", though literally "I am to eat".