いくつ

何なん個こN5sense 1
何個 asks specifically for a count of small objects, while いくつ is a softer, more general way to ask 'how many'.
何なん歳さいN5sense 2
何歳 asks age directly and can sound blunt, whereas いくつ is a gentler way to ask someone's age, often used with children.

Don't confuse with

いくらN5
幾ら sounds similar to 幾つ but asks about price or amount, not a countable number or age.
幾いく
幾つ is a standalone question word meaning 'how many' or 'how old', while 幾 is the bound prefix used inside forms such as 幾日, 幾度, and 幾久しい.
幾いく日にちN1
幾日 asks specifically about a number of days or a date, while 幾つ asks about a general quantity or age, though both share the 幾 prefix and can be confused by learners.
一ひとつN5
幾つ asks 'how many' using the same native counting series as 一つ, so beginners sometimes confuse the question word with the number itself.

Word family

いくつか
a few; some; several
いくつも
many; much; plenty · hardly any; not many
いくつでも
a great many; ever so many · any number; as many as you like; no matter how many

Etymology

Derived from 幾 (iku, 'how many') + つ (tsu, counter suffix). The exact historical development is straightforward; the word has been used for counting and age inquiries since Old Japanese.