A character can only benefit from the magical qualities of items donned in the appropriate manner. The total number is not actually limited, but there are limits.
"A character can't normally wear more than one pair of footwear, one pair of gloves or gauntlets, one pair of bracers, one suit of armor, one item of head wear, and one cloak." (DMG, pg. 141, col. 1, para. 2.)
This entry continues with a caveat for the DM only to decide: "You [the DM] can make exceptions; a character might be able to wear a circlet under a helmet, for example, or be able to layer two cloaks." Therefore, ask your DM.
Further instructions state that: "Items that come in pairs - such as boots, bracers, gauntlets, and gloves - impart their benefits only if both items of the pair are worn." (DMG, pg. 141, col. 1, para. 3.)
Magical items must be donned in the intended fashion (DMG, pg. 140, col. 2, para. 3) including rings to be placed on a finger, digit (DMG, pg. 139, col. 2, para. 2), or tentacle (DMG, pg. 141, col. 1, para. 2).
The more powerful items (rings and otherwise) have an "attunement" requirement (DMG, pg. 138, col. 2, para. 1) which, among other details, limits the magical benefits to 3 such items. The character may wear/use more than 3 items requiring attunement, but can only be attuned to 3 at a time, and will only gain the magical benefits from those 3 attuned items (plus any other magical items not requiring attunement). An attunement process prevents switching attunements on the fly (DMG, pg. 138, col. 1, para. 3).
Given that, you could pile rings onto your fingers. However, don't be surprised if your DM then rules that your character risks one or more penalties to Dexterity, attacks, increased fumble, move silently, etc., when using such laden hands.