High-quality observations of the real world are crucial for a variety of applications, including producing 3D printed replicas of small-scale scenes and conducting inspections of large-scale infrastructure. These 3D observations are commonly obtained by combining multiple sensor measurements from different views. Guiding the selection of suitable views is known as the NBV planning problem. Most NBV approaches reason about measurements using rigid data structures (e.g., surface meshes or voxel grids). This simplifies next best view selection but can be computationally expensive, reduces real-world fidelity, and couples the selection of a next best view with the final data processing. This paper presents the Surface Edge Explorer, a NBV approach that selects new observations directly from previous sensor measurements without requiring rigid data structures. SEE uses measurement density to propose next best views that increase coverage of insufficiently observed surfaces while avoiding potential occlusions. Statistical results from simulated experiments show that SEE can attain similar or better surface coverage with less observation time and travel distance than evaluated volumetric approaches on both small- and large-scale scenes. Real-world experiments demonstrate SEE autonomously observing a deer statue using a 3D sensor affixed to a robotic arm.