Winglets provide a lift boost that would be useful to a plane of this size. But the An-255 does not have any winglets.
Why is it so?
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.
Winglets were just beginning flight tests at NASA with a modified KC-135 when the AN-225 was designed. No one at the Antonov Design Bureau could have known at the time the benefits of using them.