Is it possible to expand the datagridview cell height as the user is typing on run time.

First you have to set particular cell - WrapMode property should be true.
Then you can use Paint function for datagridview.
this function automatically increased row height whatever you typed in cell

This is one of sample paint function.

protected override void Paint(Graphics graphics, Rectangle clipBounds, Rectangle cellBounds, int rowIndex, DataGridViewElementStates cellState, object value, object formattedValue, string errorText, DataGridViewCellStyle cellStyle, DataGridViewAdvancedBorderStyle advancedBorderStyle, DataGridViewPaintParts paintParts)
{
try
{
int mergeindex = ColumnIndex - m_nLeftColumn;
int i;
int nWidth;
int nWidthLeft;
string strText;
Pen pen = new Pen(Brushes.Black);
// Draw the background
graphics.FillRectangle(new SolidBrush(SystemColors.Control), cellBounds);
// Draw the separator for rows
graphics.DrawLine(new Pen(new SolidBrush(SystemColors.ControlDark)), cellBounds.Left, cellBounds.Bottom - 1, cellBounds.Right, cellBounds.Bottom - 1);
// Draw the right vertical line for the cell
if (ColumnIndex == m_nRightColumn)
graphics.DrawLine(new Pen(new SolidBrush(SystemColors.ControlDark)), cellBounds.Right - 1, cellBounds.Top, cellBounds.Right - 1, cellBounds.Bottom);
// Draw the text
RectangleF rectDest = RectangleF.Empty;
StringFormat sf = new StringFormat();
sf.Alignment = StringAlignment.Center;
sf.LineAlignment = StringAlignment.Center;
sf.Trimming = StringTrimming.EllipsisCharacter;
// Determine the total width of the merged cell
nWidth = 0;
for (i = m_nLeftColumn; i <= m_nRightColumn; i++)
nWidth += this.OwningRow.Cells[i].Size.Width;
// Determine the width before the current cell.
nWidthLeft = 0;
for (i = m_nLeftColumn; i < ColumnIndex; i++)
nWidthLeft += this.OwningRow.Cells[i].Size.Width;
// Retrieve the text to be displayed
strText = this.OwningRow.Cells[m_nLeftColumn].Value.ToString();
rectDest = new RectangleF(cellBounds.Left - nWidthLeft, cellBounds.Top, nWidth, cellBounds.Height);
graphics.DrawString(strText, new Font("Arial", 10, FontStyle.Regular), Brushes.Black, rectDest, sf);
}
catch (Exception ex)
{
Trace.WriteLine(ex.ToString());
}
}