Which ND filter do I need for a 5-minute daylight exposure?

We can actually work this out using the information given to us by the metadata and the photo itself. This is going to be a rough science, but good enough to help choose a filter.

Firstly, we have the exposure information in the photo's metadata. This tells us it was taken at f/22 for 105 seconds at ISO 100. Secondly, what does the photo itself tell us about the prevailing light conditions? The sky clearly shows it was cloudy, but the soft shadow under the bench indicates there was some sunshine too (as do the streaks in the sky, in fact). So let's call it partly overcast.

The Sunny 16 rule tells us that on a partly overcast day, at ISO 100, we'd use an exposure of 1/100s at f/11. But the photographer has used f/22, so we need to adjust the shutter speed accordingly to 1/25s (we've lost two stops on the aperture so need to add two to the shutter speed).

So, without a filter we think this scene could have been captured with something like f/22 at 1/25s and ISO 100. We're already making lots of assumptions here - it might have been a much brighter or cloudier day than we're assuming, and of course the image may have been substantially post-processed - but let's assume we're near enough, say within a stop or two. To convert that into a long exposure you'll need a neutral density (ND) filter.

So, the question now becomes: what strength of ND filter is needed to convert a shutter speed of 1/25s into one of 105s? That's somewhere around 11 stops of difference (11 stops is equivalent to multiplying the shutter speed by 2048: here it's multiplied by nearer 2600, but only according to our back-of-an-envelope calculations). If our figures are more or less correct, 11 stops of filtering would produce roughly an 80s exposure. A 10-stop filter would produce a 40-second exposure. Both of those would produce a very similar looking result to the photo you've linked to. Plus I always find very dark ND filters will take quite a bit of extra exposure: those shutter speeds may end up nearer to 100s and 50s respectively.

I'm not sure you can actually buy an 11-stop filter but 10-stop filters are common. So my answer is: you'd need a 10-stop (aka NDx1000) neutral density filter. You may want to stack this with a lighter filter (e.g. a 2 or 3-stop ND) if you really want to get extreme long exposure times.