Why does the Antonov AN-225 not have any winglets?
Winglets are less effective at producing added lift than an equal-length wingspan extension. They are used on airliners when simply enlarging the wing would put the plane into a larger size class, or outside the 80 m "box". A longer wingspan, which the An-225 has (10% over the limit), is better at producing lift. 

The An-225 is a special purpose cargo aircraft for delivering very large loads such as the Buran space shuttle. It doesn't have to care for commercial airport traffic patterns meant for high-throughput operations - servicing thousands of planes separated into fixed size groups. 

In its original role the An-225 would operate from airstrips purpose-built for it, and seeing little other traffic. Today, on the rare (or not so rare, but its most public deliveries are to remote destinations) occasion when it has a commercial airport in its itinerary, they can deal with the extra requirements of a somewhat larger plane.   

Think of it like a wide load truck: they can go outside the 102"+3" limit at the expense of pre-planning the route. It's always a special delivery with these planes.