Intelligence is multifactorial and defined in a number of ways. Psychometrics generally defines intelligence as abstract thought, reasoning and problem solving, however, the field also recognizes that experience plays a role.
For example, in standardized tests of intelligence or IQ (ie, WISC, WAIS, WAIS-R, Stanford Binet, Raven's etc.), examinees can be penalized if they have different cultural experiences than what the test is designed to measure. A former question on the WISC (which I took as a child and remember quite clearly), asked one to discuss the various definitions of the word orange. If one had never seen the fruit, they would be penalized. That would be about as fair as showing me a picture of a cherimoya and asking me to describe what it tastes like.
More recent exams have started to include measures of visual processing, reaction times, spatial ability, fluid intelligence and even musical, kinesthetic or social intelligence. These are popular because it allows for a broader definition of intelligence than the original psychometric measures allowed for and to show that different talents can be measured and fit within our definition of "intelligence". Are Yo Yo Ma, Itzhak Perlman or Michael Phelps intelligent? Perhaps not on standardized tests but certainly their talents reach levels that many of us could never hope to obtain. Practice? Yes, but there is innate talent and physical attributes (in Phelps' case) that are paramount.
So while it is true that some intelligence can be gained through experience and life, "geniuses" are not defined as such because of that but rather because of some innate quality that is well recognized, well tested and can accurately predict achievement and performance. It is clear that these individuals are born this way.
. The 8 I got on my actual test was actually my highest score 
