It's a condition in which one suffers the inability to remember to the word he or she wants to use and then becomes obsessed with trying to remember it.
What is this interesting word's root? Could someone tell me?
It sounds like a purpose-created, pseudo-medical neologism to me, like pneumonoultramicroscopicsilicovolcanoconiosis— except there aren't any results at all in CHAE or Google Ngrams. Elsewhere on the web I can only find it in lists of too-clever words, and not in any online dictionaries.
The etymology, I think, is reasonably clear:
log- This is from the λόγος (logos), Greek for word or speech.
-anamn- This comes, via Latin, from the Greek ἀνάμνησις (anamnesis), for remembering or recollection.
-osis This is a suffix denoting a state or condition, especially a disorder or abnormality, and also traceable to Greek via Latin.
Hence loganamnosis, a psychotic obsession about remembering words. Considering its obscurity, I wouldn't use it myself, especially considering there is a much better-attested word available:
onomatomania n. An abnormal concentration on certain words and their supposed significance or on the effort to recall a particular word. [Stedman's]
Loganamnosis dates from at least 1958, as listed in Reversicon: A Medical Word Finder by Jacob Edward Schmidt:
Preoccupation, morbid, with attempt to recall forgotten name . . . . NOMANAMNOSIS
Preoccupation, morbid, with attempt to recall forgotten word . . . . LOGANAMNOSIS