It is technically possible to notate in the way you describe, but it would probably not be a good idea. Time signatures are not only about cutting up the music into manageable chunks, but they also tend to guide things like stress and phrasing.
Most music isn't simply made up of a sequence of notes played robotically one after the other, but is rather made up of phrases; small musical "sentences" if you will. Ideally you want to write music down in a way which lines up nicely with how you intend the music to feel when performed.
If I; for example stuck - random bits of punctionation! into the middle of? a sentence (it would) become harder to understand what I'm: trying to convey. Just like punctuation should line up with the meaning of a sentence, so the time signature should line up with the "meaning" of a piece of music.
One place where this matters a lot is in regards to rhythm; as Michael Curtis mentions in his answer, time signatures generally determine which beats are felt as stressed or emphasized. In 4/4 the first beat is generally the "heaviest" followed by a secondary stress on beat three. In 3/4 there is a strong beat one followed by two relatively "weak" beats. 6/8 time is generally felt in terms of two beats, each divided into three etc. etc.
The "strong" or "heavy" beats receive more emphasis, and they are usually the points where we see chord changes and such (although this can of course be subverted for effect).
Thus, while you could notate almost any music in almost any time signature, you really want to notate it so that it best represents the kind of "feel" you're after in the music. Thus, if a piece is best felt in terms of five beats, then that's how it should be notated, even if it seems a bit awkward at first.