Wraps Texture2Ds around the screen by using a set of RenderTarget2Ds.

ScreenWrapper.cs

public class Game1 : Game
    {
        private readonly GraphicsDeviceManager graphics;
        private SpriteBatch spriteBatch;

        private const int _ScreenWidth = 800;
        private const int _ScreenHeight = 600;
        private RenderTarget2D _WrapTarget;

        private Texture2D _Circle, _Square;
        private Vector2 _CirclePosition, _SquarePosition;

        public Game1()
        {
            graphics = new GraphicsDeviceManager(this);
            Content.RootDirectory = "Content";
        }

        protected override void Initialize()
        {
            graphics.PreferredBackBufferWidth = _ScreenWidth;
            graphics.PreferredBackBufferHeight = _ScreenHeight;
            graphics.ApplyChanges();
            
            base.Initialize();
            
            _CirclePosition = new Vector2(_ScreenWidth / 2 - _Circle.Width / 2, _ScreenHeight / 2 - _Circle.Height / 2);
            _SquarePosition = new Vector2(_ScreenWidth / 2 - _Square.Width / 2, _ScreenHeight / 2 + _Square.Height / 2);
        }

        protected override void LoadContent()
        {
            spriteBatch = new SpriteBatch(GraphicsDevice);

            _WrapTarget = new RenderTarget2D(GraphicsDevice, _ScreenWidth * 3, _ScreenHeight * 3);

            _Circle = Content.Load<Texture2D>(@"circle");
            _Square = Content.Load<Texture2D>(@"square"); 
        }

        protected override void Update(GameTime gameTime)
        {
            if (Keyboard.GetState().IsKeyDown(Keys.D)) _CirclePosition.X += 6;
            if (Keyboard.GetState().IsKeyDown(Keys.A)) _CirclePosition.X -= 6;
            if (Keyboard.GetState().IsKeyDown(Keys.W)) _CirclePosition.Y -= 6;
            if (Keyboard.GetState().IsKeyDown(Keys.S)) _CirclePosition.Y += 6;

            if (_CirclePosition.X <= - _ScreenWidth + _Circle.Width / 2) _CirclePosition.X = _Circle.Width / 2;
            if (_CirclePosition.Y <= - _ScreenHeight + _Circle.Height / 2) _CirclePosition.Y = _Circle.Height / 2;
            if (_CirclePosition.X >= _ScreenWidth * 2 - _Circle.Width / 2) _CirclePosition.X = _ScreenWidth - _Circle.Width / 2;
            if (_CirclePosition.Y >= _ScreenHeight * 2 - _Circle.Height / 2) _CirclePosition.Y = _ScreenHeight - _Circle.Height / 2;

            base.Update(gameTime);
        }

        protected override void Draw(GameTime gameTime)
        {
            GraphicsDevice.SetRenderTarget(_WrapTarget);
            GraphicsDevice.Clear(Color.Transparent);
            
            spriteBatch.Begin(SpriteSortMode.Deferred, BlendState.AlphaBlend, null, null, null, null, Matrix.CreateTranslation(_ScreenWidth, _ScreenHeight, 0.00f));

            spriteBatch.Draw(_Circle, _CirclePosition, null, Color.White, 0.00f, new Vector2(_Circle.Width / 2.00f, _Circle.Height / 2.00f), Vector2.One, SpriteEffects.None, 0);
            spriteBatch.Draw(_Square, _SquarePosition, null, Color.White, 0.00f, new Vector2(_Square.Width / 2.00f, _Square.Height / 2.00f), Vector2.One, SpriteEffects.None, 0);

            spriteBatch.End();


            GraphicsDevice.SetRenderTarget(null);
            GraphicsDevice.Clear(Color.LightYellow);

            spriteBatch.Begin(SpriteSortMode.Deferred, BlendState.AlphaBlend);

            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(0, 0, _ScreenWidth, _ScreenHeight), Color.White);
            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(_ScreenWidth, 0, _ScreenWidth, _ScreenHeight), Color.White);
            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(_ScreenWidth * 2, 0, _ScreenWidth, _ScreenHeight), Color.White);
            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(0, _ScreenHeight, _ScreenWidth, _ScreenHeight), Color.White);
            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(_ScreenWidth, _ScreenHeight, _ScreenWidth, _ScreenHeight), Color.White);
            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(_ScreenWidth * 2, _ScreenHeight, _ScreenWidth, _ScreenHeight), Color.White);
            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(0, _ScreenHeight * 2, _ScreenWidth, _ScreenHeight), Color.White);
            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(_ScreenWidth, _ScreenHeight * 2, _ScreenWidth, _ScreenHeight), Color.White);
            spriteBatch.Draw(_WrapTarget, Vector2.Zero, new Rectangle(_ScreenWidth * 2, _ScreenHeight * 2, _ScreenWidth, _ScreenHeight), Color.White);

            spriteBatch.End();


            base.Draw(gameTime);
        }
    }