This is duly written here
docs.microsoft.com/.../upgrading-to-business-central-on-premises
For each upgrade step, you have to find out what is the latest supported upgrade version.
These should be the clean and reccomended steps to follow in your case
4.0 SP1 to 4.0 SP3 (Use the Upgrade Toolkit you find in 4.0 SP3)
4.0 SP3 to NAV 2009 R2 (Use the Upgrade Toolkit you find in NAV 2009 R2)
NAV 2009 R2 to NAV 2015 (Use the latest build, application and upgrade toolkit)
NAV 2015 to BC14 (Use the latest build, application and upgrade toolkit)
BC14 to BC15 (check the compatibility matrix between 14.x and 15.x)
In such cases, where partner and customer sticks with very very very very (I said Very? yes VERY) old versions, they should also consider to start a migration project completely instead since all technologies that do support the ERP drastically changed.
Not only the client itself and application business process but also the server machine (if not uplifting to the cloud), SQL Server version, new features, etc.
The refrain to maintain a customer always in stuck for such a long long time will always come back as a boomerang. And this customer is stuck in its position since approx. 15 years now.
It's an eternity, if you consider how fast the ERP business - and the world outside - is evolving.
