Take a byte out of it!

Jelly, 6 bytes


A monadic link taking a number and returning a number.

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Uses a nice quick, Ƥ, developed by miles...

BḄ-8ƤṀ - Link: number
B      - convert to a binary list
    Ƥ  - for loop over some slices to be determined...
  -8   - this is a negative nilad, therefore: use overlapping outfixes of length 8
       -   (exactly what the specification asks us to inspect)
 Ḅ     -   convert from a binary list to an integer (vectorises)
     Ṁ - maximum

J, 12 bytes


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          #:     to binary
     8  \.       remove consecutive groups of eight
      #.         convert each result to decimal
  >./            maximum
[:               do nothing, this lets me avoid parentheses

Python 2, 41 bytes

f=lambda n:n>>8and max(n>>8,2*f(n/2)+n%2)

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