I'm asylee, how many times can I use a valid travel document (valid for one year)? Can I use this document for one, two, three or four trips overseas while the document still valid?

What do you mean by "travel document"?

If you mean a passport they are usually valid for 5 years in Canada and I believe 8 years for the US (dont recall exactly) and I'm sure other countries in Europe would have them valid for several years also.

A passport (again if that is what you mean) does not have a limit on how many times it can be used.