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To my understanding, there are standardised kanji that have either:

  * No reliable information on how they are to be pronounced OR
  * No reliable information on what they mean OR
  * All of the above.

Do these kanji exist, and if so: What are the current hypotheses on what they
mean/how they are pronounced?
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question
kanji orthography
What are the most obscure kanji?
Yes.

幽霊文字  
<https://ja.wikipedia.org/wiki/%E5%B9%BD%E9%9C%8A%E6%96%87%E5%AD%97>

> 幽霊文字（ゆうれいもじ）とは、JIS基本漢字に含まれる、典拠不明の文字の総称。

JIS基本漢字 (aka JIS第1・第2水準漢字) is a set of Japanese characters (including approx.
6350 kanji) established in 1978. It has been widely supported by personal
computers since then, but this standard contains several kanji which have no
reliable origin.

For example, [彁](http://www.wdic.org/w/CUL/%E5%BD%81) is a kanji which
probably did not exist before the JIS基本漢字 set was established. No one knows
its meaning. It has the reading (カ/セイ), but it's a fabricated one for
convenience sake.

[戸籍統一文字](https://ja.wikipedia.org/wiki/%E6%88%B8%E7%B1%8D%E7%B5%B1%E4%B8%80%E6%96%87%E5%AD%97)
contains over 50,000 "kanji", many of which are not listed in any dictionary.
They have character codes and actually in use in government offices, but I
don't know whether you can call them "standardized".

And you can find tons of insane kanji which are _not_ standardized here:
[漢字部屋](http://www.geocities.jp/f9305710/henkanji.html)
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answer
We call them:

  * No reliable information on how they are to be pronounced → 音【おん】（未詳【みしょう】／不明【ふめい】）字【じ】
  * No reliable information on what they mean → 義【ぎ】（未詳／不明）字
  * All of the above → 音義（未詳／不明）字

幽霊文字 in @naruto's answer is usually considered as a subset of them
(technically, meaning/pronunciation of some 幽霊文字 are easily inferrable so
they're not true subset).

Some Chinese Characters have unknown meaning/pronunciation even they have
reliable source, because it's clearly written in dictionaries that
"音未詳"/"義未詳"/"音義未詳". When those massive dictionaries were compiled, they tried
to collect characters from all available documents. Some characters only have
too sporadic (or single) specimens to reasonably guess what they are. Some
others are from previous dictionary with whose description about
meaning/pronunciation are lost or unrecorded. The Unicode standard contains a
numerous number of those characters because it has unconditionally
incorporated all entries in the authoritative dictionaries.

[![enter image description
here](https://i.stack.imgur.com/Ieqgs.jpg)](https://i.stack.imgur.com/Ieqgs.jpg)

While they have possibility to be identified through investigation, there was
no research encompassing the entire encoded characters as far as I know, thus
many of them are just left to be determined.

  * [「JIS X 0208」における音義未詳字に対する原典による同定](http://dspace.wul.waseda.ac.jp/dspace/bitstream/2065/29406/1/KokugogakuKenkyuToSiryo_20_Sasahara.pdf)
  * [JIS音義未詳字の辞典](http://plaza.rakuten.co.jp/jitenfeti/7000/)
  * [諸橋轍次編『大漢和辞典』の＜音義未詳字＞](http://rose.hucc.hokudai.ac.jp/~o16404/shikeda/moro.html)
  * [《大漢和辞典》第一巻中の難字1](http://nkay1005.blog.fc2.com/blog-entry-65.html) to [大漢和辞典の未詳字23](http://nkay1005.blog.fc2.com/blog-entry-106.html)
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