diff --git a/base/intfuncs.jl b/base/intfuncs.jl
index 8d46fcffa3ad58..b84c54fb8cb3b4 100644
--- a/base/intfuncs.jl
+++ b/base/intfuncs.jl
@@ -298,7 +298,8 @@ function invmod(n::T) where {T<:BitInteger}
 end
 
 # ^ for any x supporting *
-to_power_type(x) = convert(Base._return_type(*, Tuple{typeof(x), typeof(x)}), x)
+to_power_type(x::Number) = first(promote(x, one(x), x*x))
+to_power_type(x) = x
 @noinline throw_domerr_powbysq(::Any, p) = throw(DomainError(p, LazyString(
     "Cannot raise an integer x to a negative power ", p, ".",
     "\nConvert input to float.")))
diff --git a/test/math.jl b/test/math.jl
index c0a2d8bf8c9f8f..6b225e62fcf0cb 100644
--- a/test/math.jl
+++ b/test/math.jl
@@ -1498,6 +1498,20 @@ end
     n = Int64(1024 / log2(E))
     @test E^n == Inf
     @test E^float(n) == Inf
+
+    # #55633
+    struct Issue55633 <: Number end
+    struct Issue55633Squared <: Number end
+    Base.:(*)(::Issue55633, ::Issue55633) = Issue55633Squared()
+    Base.promote_rule(::Type{Issue55633}, ::Type{Issue55633Squared}) = Int
+    Base.convert(::Type{Int}, ::Issue55633) = 3
+    Base.convert(::Type{Int}, ::Issue55633Squared) = 9
+    for x ∈ (im, pi, Issue55633())
+        p = promote(one(x), x, x*x)
+        for y ∈ 0:2
+            @test all((t -> ===(t...)), zip(x^y, p[y + 1]))
+        end
+    end
 end
 
 # Test that sqrt behaves correctly and doesn't exhibit fp80 double rounding.