The The Winged Dragon of Ra's card says that when you pay 1000 life points you can destroy a monster, but it never says anything about it being once per turn.
So can I use his effect continuously in a single turn?
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The text of The Winged Dragon of Ra reads (emphasis mine):
Cannot be Special Summoned. Requires 3 Tributes to Normal Summon (cannot be Normal Set). This card's Normal Summon cannot be negated. When Normal Summoned, other cards and effects cannot be activated. When this card is Normal Summoned: You can pay LP so that you only have 100 left; this card gains ATK and DEF equal to the amount of LP paid. You can pay 1000 LP, then target 1 monster on the field; destroy that target.
We can see that the effect of paying 1000 LP to target-destroy a monster does not have the "once per X" limitation", thus it can be used as many times as you want (that is, as many times you can spare 1000 LP) :)
Unrelated to your question (a nitpick, actually), it's not much technically correct to say "use his effect continuously", as this effect is not a continuous effect. A less ambiguous way of phrasing it could be "use his effect repeatedly"...