How to merge PDFs on an iPhone
The Files app can do this, though the steps are not obvious. Here they are, plus a faster option when you need to reorder pages.
The built-in way, through the Files app
Open the Files app, find the PDFs you want to combine, and tap Select in the top corner. Tap each PDF to select all of them, in the order you want them to appear, then tap the share icon. From the share sheet, tap Print. In the print preview, use two fingers to pinch outward on the page thumbnail; this opens a full-page preview showing every page from every selected file in one continuous document. Tap the share icon again from this view and choose Save to Files (or Save to Photos, or share it directly). What comes out is a single PDF with every page from your selected files, in the order you picked them.
Why this feels hidden
This works because the Print preview treats anything printable, including a stack of separate PDFs, as one combined document, and the pinch gesture reveals the pages behind that print job as their own file. It is a genuine, no-install way to merge PDFs, but it does not appear anywhere as a "merge" or "combine" button, and it is easy to miss the pinch-to-preview step.
The limits of this method
The Files app trick joins files in the order you tapped them, and there is no way to reorder or remove individual pages once they are in the combined preview: if you tap files in the wrong order, you have to start over. It also does not let you drop specific pages from the middle of a document before merging.
When you need more control
Stayput's merge PDF tool lets you drag files into the exact order you want, see thumbnails of every page, and remove any you do not need, before merging, all in your phone's browser. The tool sets no size limit, but your phone’s memory does, and the files are combined locally rather than uploaded anywhere, which matters for something like a signed lease or a set of ID documents.
Questions
Do I need to download an app to merge PDFs on iPhone?
No. The Files app's Print-preview trick merges PDFs natively, and a browser-based tool covers cases needing reordering, without installing anything either way.
Can I choose the order of pages when merging on iPhone?
With the Files app trick, only by selecting the files in that order beforehand. A tool with a visible page list, like Stayput's merge tool, lets you reorder afterward too.
Does merging PDFs on an iPhone upload them anywhere?
The Files app method stays entirely on the device. A browser tool that processes files locally, rather than uploading them to a server, keeps the same guarantee.
Can I merge PDFs and images together?
The Files app print trick works with PDFs and can include some image types in the same selection since both are printable. A converter that turns images into PDF pages first is the more reliable way to mix the two.