It shouldn't happen. Start over. I'm going to assume (usually a highly irresponsible thing to do), that your hanger is straight, cables are run nice and smooth, no hangups, and they are seated in the levers properly. Ok, now that we know that all those things are good... we know that right?... let's proceed as I think this is a very simple adjustment issue.
First, let's talk. Does the chain seem be over shifting a bit, as if a bit more would throw it into the spokes, or not quite shifting enough, as in it's thinking "I think I can I think I can" as it tries to climb to the larger sprocket. I ask because, despite you saying that it's shifting pretty well overall, I would question that and say it's shifting like crap, and it is, based on what you say and show. I think you're starting point is slightly off, and as it moves up the cassette, that error becomes more noticeable as it tries to climb the cassette and jump bigger gaps. I ask the question about which way it is off to ascertain if the low limit screw (the screw that limits the derailleur from going into the spokes, needs any adjusting from the get go (i.e., to loosen it off if that's what's preventing it from getting to the largest cog). But I suspect that it's the opposite and the limit screw is already loosened off enough and the chain is actually riding a little too far towards the spokes), so you can leave that screw untouched for the moment. Further reinforcing that theory is that it's having problems on the 25 cog as well, where absolutely no limit screw adjustment is at play.
The other thing I notice is that the spring on the adjusting barrel is almost, if not completely, bottomed out, making any adjustment difficult if the cable was tightened to the anchor bolt when it was quite taught to be begin with. That's important to note, because it could limit our options during adjustment to get the exact starting point.
Ok, so what is the "starting point". It's just that, with no cable attached (yes, undo it completely), make sure the upper pulley is as aligned to the smallest sprocket as possible. It's harder to do with the chain on, and I usually do this step before installing the chain. But the chain is on, so it just may take a bit more fiddling, no biggie. Also, loosen the adjuster barrel on the derailleur a couple of turns so that you actaully sill have some useable adjustment in both directions.
Ok now, loosen off the upper limit screw a bit (the one that limits the derailleur from going into the chain/seat stays). You don't want that limiting the starting point at this time and this may in fact be the entire cause of your issue, it may be too far in. But loosening it off won't affect anything if the cable is too tight and can't be adjusted for the reason I just gave concerning the adjusting barrel position. That's why we're starting from scratch and detaching the cable, as well as giving the adjusting barrel some room to breathe in both directions.
Ok, now that you've loosened off the upper limit screw, and loosened the ajdusting barrel a couple of turns, just play with the lower limit screw a bit while pedalling (in a stand), but without the rear derailleur cable attached. Loosen it until you can hear bad things starting to happen (it's too far outboard), then tigthten it until you can hear the same things start to happen on the inner side, where you can hear the chain wanting to start climbing to the second cog. Obviously that's too far in. You need to set the limit screw in the middle, or even ever so slightly outboard of the middle (and then take up the tiny slack with the adjustment barrel) of perfect alignment between the upper pulley and the smallest cog.
Now, reattach the rear derailleur cable, taking all the play out, but not so tight that you're actually actuating the derailleur with no lever movement (oh yeah, of course make sure your lever is postioned so that it would be on the smallest cog during this process).
Ok, so with cable reattached, some adjustment available in both directions via the adjustment barrel, and the starting point as close as you could get it, play with the shifting just between the smallest and second smallest cog. You should be able to go back and forth easily, and use the adjusting barrel to make sure you can. Listen for any noises that might indicate less than perfect alignment and adjust accordingly. If I nail this adjustment between these two cogs, the rest are usually perfect. Usually
.
But lets' check things out. Ok. In the stand, just shift up one by one, nice and easy, making sure each shift is perfect. If you sense it wants to overshift anywhere, then loosen the adjusting barrel a smidge, and vice versa if it's having a bit of a tough time at any stage climbing to the next cog. Back and forth, back and forth, until it's perfect. A this point, if done right, I suspect you're going to able to shift right on up to the largest cog no problem.
Now, assuming you're smiling and all proud of yourself because it seems to be working, it's time to dial in the lower limit screw. The key is to have it so that the shift from the second largest sprocket to the largest is easy and not in any way encumbered by the limit screw, but that the limit screw is just "on the limit" of it going any further. I'm going to make another assumption here and assume you know how to do that. Check how close your upper pulley is to the spoke crossings at this point, ensuring that no contact is going to happen. Once you have the limit screw set, actually try to shift into the spokes (slowly of course), just to make sure your limit screw is doing what i'ts supposed to be doing, namely, not allowing that to happen.
Ok, almost there.
You said you had the upper pulley within about 5mm of the largest cog earlier. That's good, it's where you want it. As close to the cogs as possible without the chain jamming in between the cogs and pulleys. Having it further away than necessary only does one of two things...
1) If the shifting was crisp to begin with, moving the pulley away will only make it less crisp. It'll still shift ok, but not as crisp as if it were as close as possible to the cogs. This is always the goal and why so much effort has been made by both Shimano and Campagnolo of late to improve this.
2) If the shifting was not quite perfect due to indexing being slightly off to begin with, moving the pulley away may mask some of this sloppiness by introducing some new sloppiness simply as a result of moving the pulley further away from the cogs, lulling you into a false sense that your shifting is "ok", even "perfect" maybe. I like to explain it like this... Imagine the upper pulley as close to the cogs as possible... and there is one little tiny shift gremlin sitting in his easy chair reading a book at his leisure, right up close there. When he gets the signal to shift, all he does is reach his hand over and lightly touch the pulley and bam, because of it's close proximity to the cogs, pickup is quick and easy and it shifts. Easy peasy. But now imagine the pulley way far away from the cogs. The chain has to be lifted up then dragged across the gap until it catches the pickup. It takes at least 3 or 4 of those little shift gremlins (yes I've counted) to do the same job as just the one sitting in his easy chair up nice and close. Plus, those 3 or 4 are constantly running back and forth dragging that heavy chain... it's just no fun. Treat your shift gremlins nice, and they'll return the favor.
That's it. Of course I've made quite a few assumptions right at the beginning of my post, and that's always a caveat, but if things are good there, and you follow what I've just written, you should be up and running in no time, and that one little shifting gremlin can relax and the others can go home until they're really needed, hopefully never.
Cheers!
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