And no, not one of the fillable PDFs that some places provide - though they can be slow as molasses, they’re pretty useful.
I mean a regular old PDF.
I could convert it to MS Word using one of the free tools out there.
But what I had a vague idea of doing was a way to have it brought up as sort of “page background” or watermark, and be able to place text boxes as an overlay of sorts.
Has anyone ever heard of anything like this?
I’ve done something similar by uploading to http://www.pdfescape.com/ and using a textbox tool to add to it. That was actually how I filled out my application forms for the Odyssey workshop so they were legible. 
If you have the full version of Acrobat (not just the Reader) you can edit text with varying degrees of success. It used to be called the Touch-up Text Tool; in recent versions I think there’s a more comprehensive “Content Editing” panel.
I’m assuming that you don’t have this, though, or you’d probably already know that.
ETA: just noticed that you seem to be talking about adding new text fields rather than editing existing text, in which case disregard what I just said.
Tools --> Comment & Markup --> Text Box Tool
Alternatively, save it in a photo format and set it as the background in a Word document. Format it however you like in Word.
Foxit Reader has a typewriter tool that allows you to open a text box on the document and enter information into it. I’ve used it whenever dealing with pdf forms that don’t have embedded form fields since I got my handwriting from my father (on whom be peace) — a doctor, and he wrote like one.
Can you describe a little more exactly what you want to accomplish?
And what software you’re using? I was assuming you had Acrobat, but maybe you don’t.
On a Mac, the built-in Preview app allows this. You can annotate with text or do things like drawing ovals or rectangles around parts of the document. It can also capture and save a signature using the computer’s camera that can then be used for signing documents electronically. There are also commercial tools on the Mac, like PDFPen, that allow more extensive manipulation.
We have just regular Acrobat Reader (or Adobe Reader, whatever it’s called these days).
The form in question right now is a job application, but I can see numerous other instances where this might be useful.
Green Bean: Tools / Comment doesn’t seem to have a text box tool, just stickys and highlighting. The idea of saving it as a photo is more like the vague idea I had.
OttoDaFe: Thanks - will have to check that option out.
I did use a website to convert it to Word, but the friend who needed it was having trouble working with the document anyway.
Why won’t the usual form features of Acrobat work for you? You can define text areas for any kind of content.
Aha - that seems to be something the original author has to enable:
Seconded!
Plus – it includes a certified signature feature, so you can use it for documents requiring a signature without having to print, sign, and scan.
I have Adobe Reader and I’m able to type stuff into PDF’s all the time.
YES!! That worked!! Didn’t occur to me to look under the Sign menu. Thanks!!
Anything for the Mother of Frank. 
Yeah. I think JohnT was the one who showed me that, and I’m no slouch when it comes to computers. It’s really not an obvious placement.
If you care why it’s there, it’s because the “Sign” tab covers all the features of a program Adobe acquired called EchoSign. While the main purpose of the software was signing documents, it also allowed typing instead of just signing.
It also is how Adobe Acrobat has built-in Internet-to-fax abilities. EchoSign allows you to sign, annotate and fax PDFs without ever printing them.
I think you will find that the commenting tools in the free PDF-XChange Viewer (Tracker Software Products :: PDF-XChange Editor and get the free download from the box to the right), will provide you with a lot more options for typing on, and otherwise marking up, your PDFs than Acrobat reader does. As well as simple typing, you can draw boxes (containing text or otherwise) draw shapes, straight lines and freehand lines, arrows, you name it, with your choice of text, outline and fill colors. It also enables you to run an image PDF through an OCR type process, to make the text searchable and copyable. It is way better than Adobe Reader for most purposes. (I haven’t tried Foxit for a long time, but last time I did it had fewer tools and several annoying features compared to PDF-XChange.)
To use the comment tools, just show the comment and markup tool bar from the view menu, if it is not showing already.