this is not an Array{Vector{Float64}, 3}, this is a Vector{Any}, which is just an alias for Array{Any, 1}. So the important thing is that this has dimension equals 1, instead of 3.

Also, you can’t push to a 3-dimensional array, only a 1-dimensional one. You could use a vector of vectors (Vector{Vector{Float64}}) and push to it just fine. If all of your entries are length 3, you might also want to look into StaticArrays.jl.
If you specifically want a Matrix (which is an alias for Array{T, 2}), and you also need it to be resize-able, you could try ElasticArrays.jl.

julia> Base.@kwdef mutable struct R
value::Vector{Vector{Float64}} = []
end
R
julia> push!(R.value, [5;6;7])
ERROR: type DataType has no field value
Stacktrace:
[1] getproperty(x::Type, f::Symbol)
@ Base .\Base.jl:28
[2] top-level scope
@ REPL[1]:1