Program:

// HugeNumber.java
public class HugeNumber
{
	private class DoublyNode
	{
		private int digit;
		private DoublyNode previous;
		private DoublyNode next;

		public DoublyNode()
		{
			digit = 0;
			previous = null;
			next = null;
		}

		public DoublyNode(int newDigit, DoublyNode newPrevious,
				                                 DoublyNode newNext)
		{
			digit = newDigit;
			previous = newPrevious;
			next = newNext;
		}
	}

	private DoublyNode head;

	public HugeNumber()
	{
		head = null;
	}

	public HugeNumber(HugeNumber other)
	{
		HugeNumber copy = new HugeNumber();

		if(other.head == null)
		{
			this.head = null;
		}
		else
		{
			copy.head = new DoublyNode(other.head.digit, null, other.head.next);
			DoublyNode currNode = copy.head.next;
			DoublyNode prevNode = copy.head;

			while(currNode != null)
			{
				DoublyNode newNode = 
                        new DoublyNode(currNode.digit, prevNode, currNode.next);
				prevNode.next = newNode;
				prevNode = newNode;
				currNode = currNode.next;
			}

			this.head = copy.head;
		}
	}

	public void addMostSignificantDigit(int newDigit)
	{
		DoublyNode newNode = new DoublyNode(newDigit, null, head);

		if(head != null)
		{
			head.previous = newNode;
		}

		head = newNode;
	}
	
	public HugeNumber addNumber(HugeNumber other)
	{
		if(head == null)
			return other;
		
		if(other.head == null)
			return this;
		
		HugeNumber tempNum = new HugeNumber();
		
		DoublyNode curr1 = head;
		DoublyNode curr2 = other.head;
				
		int carry = 0;
		int sum = 0;
		while(curr1 != null && curr2 != null)
		{
			sum = curr1.digit + curr2.digit + carry;
			
			if(sum < 10)
				carry = 0;
			else
			{
				sum = sum % 10;
				carry = 1;
			}
			
			tempNum.addMostSignificantDigit(sum);
			
			curr1 = curr1.next;
			curr2 = curr2.next;
		}
		
		while(curr1 != null)
		{
			sum = curr1.digit + carry;
			
			if(sum < 10)
				carry = 0;
			else
			{
				sum = sum % 10;
				carry = 1;
			}
			
			tempNum.addMostSignificantDigit(sum);
			
			curr1 = curr1.next;
		}
		
		while(curr2 != null)
		{
			sum = curr2.digit + carry;
			
			if(sum < 10)
				carry = 0;
			else
			{
				sum = sum % 10;
				carry = 1;
			}
			
			tempNum.addMostSignificantDigit(sum);
			
			curr2 = curr2.next;
		}
		
		if(carry > 0)
		{
			tempNum.addMostSignificantDigit(carry);
		}
		
		HugeNumber hugeSum = new HugeNumber();
		DoublyNode current = tempNum.head;
		
		while(current != null)
		{
			hugeSum.addMostSignificantDigit(current.digit);
			current = current.next;
		}		
		
		return hugeSum;
	}

	public void reset()
	{
		head = null;
	}

	public String getNumber()
	{
		return this.toString();
	}

	public int size()
	{
		int count = 0;
		DoublyNode current = head;

		while(current != null)
		{
			count++;
			current = current.next;
		}

		return count;
	}

	public String toString()
	{
		String result = "";

		DoublyNode current = head;

		while(current != null)
		{
			result = current.digit + result;
			current = current.next;
		}

		return result;
	}
}

// HugeNumberDemo.java
public class HugeNumberDemo
{
	public static void main(String[] args)
	{
		HugeNumber number1 = new HugeNumber();
		
		number1.addMostSignificantDigit(9);
		number1.addMostSignificantDigit(8);
		number1.addMostSignificantDigit(7);
		number1.addMostSignificantDigit(6);
		number1.addMostSignificantDigit(5);
		number1.addMostSignificantDigit(4);
		number1.addMostSignificantDigit(3);
		number1.addMostSignificantDigit(2);
		number1.addMostSignificantDigit(1);
		number1.addMostSignificantDigit(0);
		number1.addMostSignificantDigit(9);
		number1.addMostSignificantDigit(8);
		
		System.out.println("Number1: " + number1.getNumber());
		
		HugeNumber number2 = new HugeNumber(number1);
		System.out.println("Number2: " + number2.getNumber());
				
		HugeNumber number3 = number1.addNumber(number2);
		System.out.println("\n" + number1 + " + " + number2 + " = " + number3);
	}
}

Output:

Number1: 987654321098
Number2: 987654321098

987654321098 + 987654321098 = 1975308642196