how precisely vague do you want to be?

Estimates are so popular in software development. You know they don't mean much. But those who ask for them defend them fervently and even make good arguments in their favour. Such people are the only ones who reap any benefit from them. These estimates helped them feel more at meetings where they always felt on the spot, because these people often knew even less than you did about what was to be done.

And then someone came up with guesstimate. It's a punny portmanteau and may have become more common, because it added colour to statements that had otherwise become bland. Ears perked up when they heard this word instead. Nothing changed. Things arguably just got obviously worse. When someone wanted a guesstimate instead of an estimate, who could blame you for taking the hint that you were supposed to just throw out a number. High numbers were (and still are) usually scare people, especially when you are talking about how long it would take you to finish something.

This is not to deny that guesstimate does not offer something that estimate does not. A guesstimate is an estimate "based on guesswork or conjecture" (source), it is an estimate that you would provide when you lacked enough information. Statisticians (who coined this new term) would appreciate the distinction, but corporate managers appreciated the sound of the word. The meaningful distinction was lost, because an estimate as practised in management often lacked enough information to begin with. A guesstimate was thus a sassy synonym.

Inevitably, the American propensity to derive a metaphor from the world of sports (preferrably baseball or basketball, both of which are popular in America) gave us ballpark estimate. This likely came from the use of a ball park or baseball stadium to convey a sense of an "acceptable range of approximation" (that and other theories are discussed over here).

It would not be unreasonable to expect to soon hear people asking for ballpark guesstimates. Had there been a larger popular area than the baseball stadium, we would have seen phrases based on it.

There is a lot of good advice about dealing with such requests. There's also good material for laughs and that same source also provides remarkably insightful explanations for useful things like Fermi approximations. Hmm. Now that's something to try out the next time there's a meeting to discussing and trading numbers.