Can adding power storing capacitors burn up an adapter?
You will have no problem with only 300uF of capacitance.  Although the comments by other posters are true you won't see any problems at this level.

300uF from a 2Amp supply will meant it will only take ~2ms to charge up to 5V.  The amount of energy involved will not cause any trouble.

The comment about conduction angle for the rectifier is true but there is invariably enough resistance and leakage inductance in the transformer and series resistance in the capacitor so that it is not a problem in low power supplies - if you were designing a supply for many tens of Amps then other configurations would need to be considered.