We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. This paper improves several previous partial characterizations and has a dual interest: - Concerning recursive analysis, our results provide %fully machine-in­de­pen­dent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type $2$) Turing machines. - Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers and provide new insights for understanding the relations between several analog computational models.