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Theoretically gravity is strongest at the centre of the Earth due to the gravitational equation. However the strength of gravity is very different from the strength of the effect. For example if you were at the centre gravity would be accelerating you the fastest, how ever this would be in all directions, so the overall effect of gravity would be 0 m/s^-2. But the place you would weigh the most I would assume is where there's a balance between distance from the centre and furthest of negative gravity pushing you up.
Alf - not sure you're right there. Although gravity varies inversely with r^2 - which might indicate that it should increase as you go down - the amount of matter attracting you to the centre also goes down. The shell theorem tells us that matter further from the centre of mass than we are will not have an effect so although the force increases proportional to the second power as r decreases - the amount of mass attracting decreases with the volume inside us proportional to the third power. In all force will is proportional to r when inside the sphere - so as r decreases so does the force.