You need Acrobat Pro for that, not just Acrobat Standard, which is what I
assume you have.
You can use the Open action of the first page (go to the Pages panel, right-click the first page, select Page Properties, select the Actions tab), but that would require you tweaking your code so the field doesn't get filled-in each time the user browses to that page.
Thank you for the Page Properties hint! Works excellent on single page files!
Are you able to explain how the doc-level script has to look like, to add my small little script?
var f = this.getField("DateFilled");
f.value = util.printd("mmm dd, yyyy", new Date());
How will it be possible to alter it afterwards?
Remember, if you figure out how to get it in, you WON'T be able to get it out. One day you may want to change it, so be sure to keep the original PDFs without the script.
You can run this code from the console to apply the script to the file:
this.addScript("Scripts", "var f = this.getField(\"DateFilled\"); f.value = util.printd(\"mmm dd, yyyy\", new Date());");