Problems in $\mathsf{BPP}$ not known to be in $\mathsf P$?
If you're asking for independent problems, how about:


  Find a prime in the interval $[N, 5N/4]$,
  Find two primes whose product is in the interval $[N, 9N/8]$,
  Find three primes whose product is in the interval $[N, 17N/16]$,
  Find four primes whose product is in the interval $[N, 33N/32]$,
  Find five primes whose product is in the interval $[N, 65N/64]$,
  $\ldots$. 


It's overwhelmingly likely that if you actually had a polynomial algorithm to solve the first of these, you would have a polynomial algorithm for all of them. But I don't see how to formally reduce any of these to any of the others. Of course, the problem


  Find a prime in the interval $[N, N+\log^{17} N ]$


solves all of these.