I am learning Japanese, and need to enter romaji into my keyboard to enter Japanese into documents and tables on my smartphone.
My textbook shows ī as the only romaji that can be spelled as a double ii.
Is this a commonplace romaji writing convention, and if so, then why is this?
EDIT:
Would I be right in saying that it is because the alphabetical combinations いい occur in Japanese (words), whereas ああ, うう, ええ, おお do not?
Hepburn originally transcribed いい as ī, but in later editions he changed it to ii. Unfortunately, his book doesn't explain why he made this decision, so I can't say with certainty what the reason was.
I think one reason is most obvious and likely, however: ī looks very similar to i, and writing ii avoids confusing the two. This isn't a problem for a, u, e, or o because they don't normally have dots over them, so they're visually distinct from ā, ū, ē, ō.
Note that Hepburn did not make the distinction mentioned in Eiríkr Útlendi's comment; he transcribed いい in all cases as ii. Although a few people do make that distinction, it's not common to do so, and you don't have to worry about it as a learner.
You wrote:
Would I be right in saying that it is because the alphabetical combinations いい occur in Japanese (words), whereas ああ, うう, ええ, おお do not?
No. We can find all five kana sequences in Japanese words. The easiest examples for the rest are interjections: ああ, ううん, ええ, and おお. Interjections aside, ああ is also an adverb, and おお appears in a number of words such as おおい and おおげさ. We can find more examples if we include sequences that cross morpheme boundaries, such as ええん (会厭) and ふうう (風雨).
If you intended to include examples where a long vowel follows a consonant (although these are technically excluded by what you wrote), we can find more examples rather easily: おかあさん, ずうずうしい, おねえさん, and ほのお.
The explanation in your edit doesn't really make sense.
There are several kinds of romanizations, each with its own set of rules.
Those rules were decided by whoever designed that romanization.
So if you ask the reasons for a certain romanization... who knows? Probably only the creator of that romanization knows the reason.
But I have a guess: I think ii is listed just because i and ī are too similar and difficult to tell apart.
So in order to avoid confusion they may use ii instead of ī.
( Maybe someone said this in the comment already. )
I think it's because (the long vowel) いい appears in regular words, as -- かわいい、いいこと、いいます、（人名、地名： いいだ） . . .
( So if you romanize these, they'd be: kawaii, iikoto, iimasu, iida, ... )
Whereas ああ, うう, ええ, おお usually do not. Instead, they are usually written with the bar (?) "ー" （長音記号） sign :
アーサー、ツアー、アームストロング、（嗚呼） 『あゝ玉杯に花うけて』
ウーロン茶、ウーパールーパー
エールフランス、エーデルワイス -- Exceptions where ええ is used : 「ええ話や…」
オープン、オーストリア、オート三輪
おお is used : おおかみ、オオカミ、オオクワガタ、… おおきな、おおらかな、おおしい、（人名、地名： おおき, etc.）
But "ee" and "oo" are avoided because they may be confusing : "feet" and "food", etc.
the double ii was copied form european laguages ( romanian an italian).. as this did not exist in Japanese languages before 19 century and japan look towards europe for inspiration. It could also be that japan has strong feelings towards(hawaii)
