I know around 10,000 Japanese words. As far as kanji goes, I've studied some 1200 of them intensively; through vocab I know many more as parts of words. I quite often have to reach for a dictionary when reading.
If you know 777 kanji in some way (like associating them with meanings, through your native language) and you haven't crammed on any vocabulary, you absolutely will not be able to read a thing.
In fact, even if you continue that way and memorize over 2000 kanji, and recognize every single one in a given document, you still won't be able to read anything without vocab.
The broadened knowledge will help support vocabulary building, though.
> I quite often have to reach for a dictionary when reading
So when reading kanji, how do you look up a word (picture) you don't know? Since there isn't a minimal set of characters, the notion of "alphabetical order" seems impossible. Weird that I've never thought about this until now, but I'm honestly baffled.
Let's say I see the word: 銀行 and want to look up the first character. I notice that the left hand side is 金 so I go to a dictionary, turn to the kanji radical section[1] find 金 and then find the original character by number of strokes[2]. This character has 14 strokes. If you have any experience with kanji then counting strokes is pretty trivial.
Also note that on the site linked to common kanji have a red background. Many of the characters are obscure so radical + stroke count narrows done the choices to very few kanji.
These days, if I'm reading print, I use the kanji appendix of the dictionary I'm using (big red Kokugo Jiten, by Kodan-sha, 1993). The kanji are grouped by stroke count, then within stroke count by radical. Usually kanji lookup can be avoided.
Firstly, unusual words tend to have furigana, which makes it trivial. Next, if the unknown kanji isn't the first one in the word, it's possible to do a prefix-based dictionary lookup. In many cases, also, I've been able to guess a reading by common structure. E.g. both 赤(red) and 跡(traces, remains) have a "seki" reading due to a common element, and 根(root) and 痕 (traces, remains) have a "kon" reading, also due to a common element. You might be able to guess at 痕跡 (konseki) by thinking of 根赤.
How we can find 跡 in the Kokuko Jiten's appendix is by counting the strokes first: 13. Then in the 13 stroke section, of the appendix, we find the subsequence of 13-stroke kanji that have the 足 seven stroke radical. The radicals are sorted by stroke count also, so we can find this subsequence fairly quick.
I can recognize quite a number of words that have at least one kanji which doesn't occur in any other word that I know. I've never studied the kanji in isolation, but I can recognize it in that word.
If you're native or an advanced learner, you probably have a good guess at the pronunciation too, so you can just type it phonetically and have your phone/computer convert for you.
The "alphabetical" order is based on the sound of the word. If you are trying to look up a kanji you don't know in a traditional dictionary, you generally look it up by the radicals (parts) of the kanji or the number of strokes of the character, but most people these days use an electronic dictionary where you draw the kanji to look it up.
If you know 777 kanji in some way (like associating them with meanings, through your native language) and you haven't crammed on any vocabulary, you absolutely will not be able to read a thing.
In fact, even if you continue that way and memorize over 2000 kanji, and recognize every single one in a given document, you still won't be able to read anything without vocab.
The broadened knowledge will help support vocabulary building, though.