Scandinavia was one of the few European areas that was never in control of the Roman Empire. What did Scandinavian civilizations think of the Roman Empire at its greatest extent?

Scandinavians were generally quite aware of the empire. Their attitude is a mix of emphasis on how different their societies are, and admiration and emulation of Roman practices. Runic writing, for example, was developed based on Roman script, but Roman script itself was not simply copied. Similarly, the Roman emperor appears in the legend of Sigurd, but here he appears as a warlord equivalent to Atilla the Hun, or the more familiar kings of the Goths or the Burgundians. Many upper-class Scandinavians went through a military career in the empire, through which they picked up elements of Roman culture (like drinking wine, or whitewashing walls) that they brought over, but the fundamental architecture of drinking halls or homes were still traditionally Scandinavian. In a sense, Scandinavians tried to mirror, and for their own purposes distort, the image of the empire to their peers.

Some nice books are, for Denmark: The Spoils of Victory. The North in the Shadow of the Roman Empire. Edited by L. Jørgensen, B. Storgaard and L.G. Thomsen. Nationalmuseet, Copenhagen, 2003

And for Sweden: Roman Reflections in Scandinavia. Eva Björklund, Malmö Museer. L'Erma di Bretschneider, 1996

Note that Klavs Randsborg's recent book 'Roman Reflections' (2015) is more about the Roman perception of Scandinavia than the other way around (but I think it's still a very nice book).