The Smurfs have one - to smurf - a generic verb that can be smurfed to almost anything.
(The Marklars don’t count, because their generic word was for nouns)
But do such things exist in any real-world languages? - closest thing I can think of in English would be to do - which can have a fairly broad range of meanings.
Latin has two do-words: facio and ago. Between the two of them, they can cover for almost any verb, but they don’t quite cover the same territory. Facio has more of a sense of making, while ago has more of a sense of dealing with something which already exists.
Same thing going on, more-or-less, with the German machen and tun. They cover for pretty much any other verb, but they overlap only slightly in a semantic sense. Since my knowledge of German is mostly intuitive (I learned it through osmosis), I can’t quite explain how to use either in any sense other than ‘it just sounds right’. Perhaps someone more linguistically-adept than myself can explain it.
There’s an aphorism by Kurt Tucholsky, a German writer and satirist: Ask an English person how to say a given German verb in English, and instantaneosly he’ll come up with a dozen different words which are absolutely synonymous. Ask a French person how to say it in French, and he’ll think about it for half an hour before finally saying “faire” (to do). “Faire” actually comes up in a lot of French idioms, and if you look it up in a dictionary, you’ll get a long list of possible meanings, depending on the context.
Tucholsky’s phrase also goes to show you what non-native speakers realize pretty quickly when learning English: English is very rich in synonyms because in addition to its Germanic roots it adopted loan words from a variety of foreign languages. On the other hand, this makes English poor in one-size-fits-all words which occur in other languages.
I’d like to add that Esperanto, with its handy system of prefixes and suffixes used to form new words out of existing ones, has a universal suffix, -um-, without a defined meaning. This suffix is used when a word is meant to be derived from another one, but there’s no other prefix or suffix that truly fits. There are examples for the usage at Wiki
If you want an example for a nearly universal noun in natural languages, you might think of res in Latin. It’s usually translated simply as “thing,” but it can mean anything from “lawsuit” to “state” to “subject matter,” and many more.