I bought my PS4 with the Destiny bundle, but I lost the disc and can't play it. Is there a way I can still play it without the disc?
Short Answer: No.
Additionally, purchasing the Digital Download version from the PS Store will simply create a second installment on the PSHD, but your account itself is unaffected. The only thing you can do is re-purchase a physical copy but don't be disheartened, if you have DLC it will remain because it's tied to your user account and already unlocked and licensed to the account, not the physical media.