The ability to enumerate objects without counting is known as subitizing.
Most studies suggest that we can subitize up to about 3 or 4 items (e.g. Starkey & Cooper, 1995). Enumeration of a small number of objects (i.e. subitizing) yields consistent response times regardless of the quantity of objects. Enumeration of larger quantities (i.e. counting) yields response times that increase linearly with the number of objects. This is taken as evidence that the two processes are computationally distinct.
Further studies in both neuropsychology (Dehaene & Cohen, 1994) and neuroimaging (Piazza et al., 2002) have supported the idea that separate neural substrates underly the two computational processes.
Dehaene, S., & Cohen, L. (1994). Dissociable mechanisms of subitizing and counting: Neuropsychological evidence from simultanagnosic patients. Journal of Experimental Psychology-Human Perception and Performance, 20(5), 958-975. PDF
Piazza, M., Mechelli, A., Butterworth, B., & Price, C. J. (2002). Are subitizing and counting implemented as separate or functionally overlapping processes?. Neuroimage, 15(2), 435-446. PDF
Starkey, P., & Cooper, R. G. (1995). The development of subitizing in young children. British Journal of Developmental Psychology, 13(4), 399-420.