Rhod, based on the refs below, I'm pretty sure that it will be the trihydrate. A quick check on Gmelin gave the following:
For the trihydrate to the anhydrous salt:King, F. E.; Partington, J. R.; Trans. Faraday Soc.; Vol. 23; (1927) 522 - 531
'complete dehydration on strong heating in vac. or in a stream of inert gas'Plotnikow, W. A.; Fialkow, J. A.; Tschalij, W. P.; Z. Phys. Chem., Abt. A; Vol. 172; (1935) 304 - 313.
'dehydration on heating 4-5h in a dry H2 stream to 200-220.degree.C'Hart, A. B.; Partington, J. R.; J. Chem. Soc.; (1943) 104 - 118.
'treatment of hydrate with a warm dry stream of N2, then slow heating to melting for some hours and then heating in vac.'For the trihydrate to the monohydrate:Bogorodski; Zh. Russ. Fiz. - Khim. O - va., Chast Khim.; Vol. 29; (1897) 182 - 182.
'heating, pptn. at 240.degree.C; sucking off with filter with sponge slab at 80.degree.C (decompn. of mother lye at 100.degree.C)'For the trihydate to the dihydrate:Same ref. as above,
'boiling trihydrate at 240-260.degree.C, crystn. on cooling to 80.degree.C; sucking off mother lye by sponge'Edit: For the dihydrate to the anhydrous salt:
Foosnaes, Trygve; Oeye, Harald A.; Acta Chem. Scand., Ser. A; Vol. 35; (1981) 81 - 90; English.
'drying at 300.degree.C and 10-8atm for 24 h; filtration, recrystn. from melt'All the above reactions are carried out under solventless conditions.
It's interesting, because I thought the solubility of the hydrate would be
far lower than that of the any of the hydrates. With lithium perchlorate I don't think the trihydrate can be used as a catalyst in organic solutions because it won't dissolve; the anhydrous form dissolves to make a 5M solution in ether whereas less than a gram of the trihydrate dissolves in 100g ether.
Maybe you could check the solubilities of your LiI vs. commercial anhydrous LiI, or check if the mass of your recrystallised LiI decreases after subjecting it to one of the methods above. That way, we'll know whether they really are overkill - I'd be happy to be wrong on this one!
Edit: The closest thing I can find to the metathesis is in: Gnessin, I. D.; Dorossinski, L. S.; Khim.-farm. Promyshlennost' (russ.); 5; (1934) 30 - 32; C. II; 1935; 1914.
However, the use of iron (II) iodide makes it impractical as this is even more expensive than LiI. If you're interested, it's the reaction of lithium carbonate with FeI
2 in water, then drying over sulfuric acid and storing in darkness. This produces the trihydrate.
More interesting ways to make LiI.3H2O:King, F. E.; Partington, J. R.; Trans. Faraday Soc.; Vol. 23; (1927) 522 - 531.
'formation by neutralization of Li2CO3 with HI soln.; no solvent given'Troost; Ann. Chim. Phys.; Vol. 51; (1857) 128 - 128.
'soln. of Li2CO3 in HI, various conditions;'It would seem there is no direct way to make the anhydrous LiI salt; all methods go via the trihydrate intermediate. Still, it'd be much cheaper to make the trihydrate with HI/lithium carbonate, and then drying it. Anhydrous LiI is rather expensive.
BTW Rhodium, it seems that LiI is happy to crystallise with almost any solvent; it crystallises with 4 moles of ethanol, so you may have inadvertantly produced LiI.4EtOH during recrystallisation. These being 'colourless plates', according to Turner; Bisset; J. Chem. Soc.; Vol. 105; (1914) 1784 - 1784.; it mentions that they are stable below 10
oC but doesn't say what happens above that.