user16577: The rose is a piece that is best described as a "circular nightrider", able to reach up to 32 other squares from the center of a 13x13 area by making consecutive knight jumps in a rotating fashion in a single move as long as no landing square en route is occupied. Incidentally, this rotating movement allows it to essentially pass a turn by returning to the square which it started its move from if it's completely unobstructed along at least 1 circle; though it obviously does not protect itself this way. I'll provide a diagram to illustrate its movement capabilities:

With 16 roses, including but still obstructed by squares occupied by other roses except the square occupied by an observed rose itself; what is the maximum number of squares on an otherwise clear 16x16 board that can be double-attacked while all 256 squares are attacked at least once each? For the purposes of this question, a square that is attacked 3 or more times does not count for anything more than a square that is attacked twice.
I feel I should note that I personally have yet to find an arrangement that attacks all 256 squares at least once by my own efforts; the very corners of the board in particular appear to require poor positioning in order to reach. However, seeing as 16 roses times 32 squares a single rose can theoretically attack creates a theoretical max of 512 attacks; I will be very surprised if an arrangement that meets the "attack each square at least once" prerequisite actually doesn't exist. If it should seem that this is the case, I would like an explanation as to why.

user8201: Sopel hasn't been around for a couple of years, but his solution is pretty and worth posting.

The pink squares contain the 16 roses. Each rose attacks all the squares whose distance away is sqrt(5), 4, 3 x sqrt(2), sqrt(29), 4 x sqrt(2) or 6. A rose does not attack the square it occupies.
All the squares are attacked from 1-3 times, including the squares which are actually occupied by the roses.
Addressing the original question: 100 squares are attacked more than once, of which just 8 are attacked three times.

16577: <p>The <a href="https://en.wikipedia.org/wiki/Chess_on_a_really_big_board" rel="nofollow noreferrer">rose</a> is a piece that is best described as a "circular <a href="https://en.wikipedia.org/wiki/Nightrider_(chess)" rel="nofollow noreferrer">nightrider</a>", able to reach up to 32 other squares from the center of a 13x13 area by making consecutive knight jumps in a rotating fashion in a single move as long as no landing square en route is occupied. Incidentally, this rotating movement allows it to essentially pass a turn by returning to the square which it started its move from if it's completely unobstructed along at least 1 circle; though it obviously does not protect itself this way. I'll provide a diagram to illustrate its movement capabilities:</p>

<p><a href="https://i.stack.imgur.com/9uB8r.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9uB8r.png" alt="A diagram illustrating the movement capabilities of a rose"></a></p>

<p>With 16 roses, including but still obstructed by squares occupied by other roses except the square occupied by an observed rose itself; what is the maximum number of squares on an otherwise clear 16x16 board that can be double-attacked while all 256 squares are attacked at least once each? For the purposes of this question, a square that is attacked 3 or more times does not count for anything more than a square that is attacked twice.</p>

<p>I feel I should note that I personally have yet to find an arrangement that attacks all 256 squares at least once by my own efforts; the very corners of the board in particular appear to require poor positioning in order to reach. However, seeing as 16 roses times 32 squares a single rose can theoretically attack creates a theoretical max of 512 attacks; I will be very surprised if an arrangement that meets the "attack each square at least once" prerequisite actually doesn't exist. If it should seem that this is the case, I would like an explanation as to why.</p>
 8201: <p>Sopel hasn't been around for a couple of years, but his solution is pretty and worth posting.</p>
<p><a href="https://i.stack.imgur.com/NRTzr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NRTzr.png" alt="enter image description here" /></a></p>
<p>The pink squares contain the 16 roses. Each rose attacks all the squares whose distance away is sqrt(5), 4, 3 x sqrt(2), sqrt(29), 4 x sqrt(2) or 6. A rose does not attack the square it occupies.</p>
<p>All the squares are attacked from 1-3 times, including the squares which are actually occupied by the roses.</p>
<p>Addressing the original question: 100 squares are attacked more than once, of which just 8 are attacked three times.</p>
