What are the greatest remaining structures the Incas built without using mortar? How could they do it?
Dry stone constructs required careful work and clever design, but most construction was done in adobe and fieldstone-and-mortar.
As to how the Incan architects could build without mortar -- they didn't. The Inca used adobe and fieldstone-and-mortar construction in most of their work -- but their dry stone construction was still spectacular.
Machu Pichu, Ollantaytambo, and other Incan ruins have become major tourist sites.
Of the Incan structures, there's three major design types at work: adobe (in which mud and clay form walls, sometimes with stone foundations), fieldstone design (essentially like brick and mortar, only using fieldstones), and dry stone design (the kind for which the Inca were most famous, despite it being used in a minority of their architecture).
Adobe and fieldstone were used in the majority of Incan architecture, and closely match up to European design strategies.
Dry stone construction, on the other hand, relied on carefully cut and polished stones (notably given rounded edges), which gave them the famous "fitted" look. By polishing the stones to uniformity, the Incas allowed stones to fit together like pieces in Tetris. The rounding of edges relieved walls made via dry stone of stress points, and gave the stones room to move under stress. In heavy earthquakes, this gave walls the appearance of dancing, before settling back into their proper alignment. Dry stone could be done with monolithic stones or small stones.
Here, you see Machu Picchu. Note the combination of large and small stone design throughout the structure. Due to the location of many Incan cities and the monumental size of their structures, both stones and stoneworkers were often imported to assist with construction.