I was certified in 1988 and kept a paper log from then till 1994 when I got my first downloadable computer. I then kept an electronic log of these dives, but for quite a few years I would print them out and store in folders. For the past decade I have not printed them out.
Currently I keep the following logs:
1 Excel spreadsheet of all my dives
2 Subsurface downloads of all dives since I got my latest computer in 2012
3 Brief description of each dive on my web site for the past (?) five years
4 Details of each dive and conditions on my web site for the current year
5 PDFs of each year's dives from Excel spreadsheet on my web site
6 PDF printouts from Subsurface for all dives since mid-2012
Why? I find it very handy to be able to refer back to a dive when I return to a spot, even locally, to see what we saw, the maximum depth etc. This helps make for a better dive.
As of today I have done 3,569 dives. All are real dives, no swimming pools, no 2 minute dips to retrieve a weight belt, although I do record short dives to deeper depths (eg releasing an anchor at 30 metres) and dives on my yacht's mooring.
Like some others have mentioned, some people who do not record their dives greatly exaggerate the number of dives they have done. I know one bloke who swore that he did 200 dives a year every year at a location near Sydney (as well as local dives). Considering that means 2 dives every day of every weekend, it was impossible for him to achieve this. Why? Because weather on average (from my records) show that at least 33% of weekends were lost to poor weather and also because I knew he was a shift worker who had to work at least 50% of the weekends each year (and there was no dive operators during the week).
When someone claims a very high number, break it down to how many this means per year and then how many per week. Unless they are a professional, then they are unlikely to be able to achieve the number they claim to have done.