With the scope of electronics being rapidly extended into emerging areas such as wearable, body-attachable, or disposable electronics, there is strong demand for realizing a flash memory - an indispensable element in modern electronic system - on various non-rigid platforms. Even with the advances achieved for flexible thin-film transistors over the last few decades, however, it has been challenging to develop a flash memory that exhibits practically viable performance and a high degree of flexibility at the same time. This setback originates mainly from the scarcity of soft dielectric materials that exhibit insulating properties that are sufficient to construct a flash memory, which involves dual dielectric layers respectively responsible for tunneling charges into a floating gate and blocking charges from leaking therefrom. Here we report ultra-flexible organic flash memories based on polymeric dielectric layers prepared by initiated chemical vapor deposition (iCVD). With the near-ideal dielectric characteristics of iCVD polymers and a device structure based on a rational design and material choice, we demonstrate a flexible flash memory bendable down to a radius of 300 $\mu$m that exhibits a projected retention over 10 years with a programming voltage on par with the present industrial standards. The proposed memory technology is then applied to non-conventional substrates such as paper to demonstrate its feasibility in a wide range of emerging applications.