The speed of light is finite. So, if we where to cover 200 light year radius we would need to expand Pi(200+r)^2 = 1.015 Pi * 200^2 = 1.5 light years in a single year which can't happen.

Put, another way 1.015^10,000 = ~10^64 but there are only ~500 billion stars in our galaxy and that's spread over 100,000 light years.

PS: The milky way only weighs ~4×10^41 kg humanity weighs ~1/2 trillion kg. So at 1.5% growth it takes less than 5,000 years for the projected mass of humanity 10^44kg to be 1,000 times as large as the mass of the milky way.