The continuous across files part is the real requirement, and it decides the approach: number after combining, not before, so the count flows naturally. In order of tooling:

The clean sequence: merge the PDFs into one first, then add page numbers to the merged document, which gives you the continuous numbering automatically since it is now one file. Adobe Acrobat does both, Combine Files then the header and footer tool with the page number token and the number spans the whole document by default. This ordering solves the continuous problem for free rather than fighting to sync separate counts.

The free desktop route: PDF tools like the Stirling PDF suite or the PDF-XChange Editor's free features merge and stamp page numbers, and a merge then number workflow in any of them produces the same continuous result. For a one time filing job these cover it without a purchase.

The scripted route for recurring or large jobs: a short Python script with pypdf merges the files, then a library like reportlab overlays page numbers onto each page, full control over position, format, Page X of Y and starting number. Worth it when this is a monthly filing rather than a one off, the script becoming a reusable tool.

The detail that matters for filing specifically: legal and formal filings often want a specific number format and position, bottom center or bottom right and sometimes Bates numbering rather than plain page numbers, which is a distinct thing covered in the Bates stamp thread linked alongside. Confirm the filing's exact requirement before numbering, since renumbering a combined document is easy but redoing a filing is not.