If you like taking the scenic route, I found a path with 17 places:
Manila -> Canada -> Havana -> Panama -> Manama -> Monaco -> Malabo -> Malawi -> Tarawa -> Ottawa -> Astana -> Tirana -> Guyana -> Guinea -> Luanda -> Rwanda -> Uganda
Although I must confess, some programming was involved.
I used wikipedia's List of national capitals in alphabetical order for the source of capitals and countries.
There's an 18 place path if you count Sana'a:
Manila -> Canada -> Havana -> Panama -> Sana'a -> Manama -> Monaco -> Malabo -> Malawi -> Tarawa -> Ottawa -> Astana -> Tirana -> Guyana -> Guinea -> Luanda -> Rwanda -> Uganda
Other interesting paths:
Longest shortest routes between two places:
11 'Bissau -> Nassau -> Maseru -> Majuro -> Malabo -> Manama -> Havana -> Guyana -> Guinea -> Luanda -> Rwanda'
11 'Bissau -> Nassau -> Maseru -> Majuro -> Malabo -> Manama -> Havana -> Guyana -> Guinea -> Luanda -> Uganda'
11 'Rwanda -> Luanda -> Guinea -> Guyana -> Havana -> Manama -> Malabo -> Majuro -> Maseru -> Nassau -> Warsaw'
11 'Uganda -> Luanda -> Guinea -> Guyana -> Havana -> Manama -> Malabo -> Majuro -> Maseru -> Nassau -> Warsaw'
Longest scenic routes between two places:
25 Russia -> Lusaka -> Luanda -> Guinea -> Guyana -> Tirana -> Astana -> Ottawa -> Tarawa -> Harare -> Havana -> Canada -> Manila -> Panama -> Sana'a -> Manama -> Malawi -> Malabo -> Maputo -> Majuro -> Maseru -> Roseau -> Bissau -> Nassau -> Warsaw
25 Rwanda -> Uganda -> Luanda -> Guinea -> Guyana -> Tirana -> Astana -> Ottawa -> Tarawa -> Harare -> Havana -> Canada -> Manila -> Panama -> Sana'a -> Manama -> Malawi -> Malabo -> Maputo -> Majuro -> Maseru -> Roseau -> Bissau -> Nassau -> Warsaw
25 Uganda -> Rwanda -> Luanda -> Guinea -> Guyana -> Tirana -> Astana -> Ottawa -> Tarawa -> Harare -> Havana -> Canada -> Manila -> Panama -> Sana'a -> Manama -> Malawi -> Malabo -> Maputo -> Majuro -> Maseru -> Roseau -> Bissau -> Nassau -> Warsaw