Best way to get the greatest date from a single row

Assume the following table...

CREATE TABLE Dummy_Data
(
ID INT,
TextField VARCHAR(20),
DateField DATE
)

INSERT INTO Dummy_Data (ID, TextField, DateField)
VALUES

   (1, 'Random Text', '2018-01-04'),
   (1, 'Random Text', '2018-02-04'),
   (1, 'Random Text', '2018-05-01'),
   (2, 'Random Text', '2018-01-14'),
   (2, 'Random Text', '2018-06-05'),
   (2, 'Random Text', '2018-01-01'),
   (2, 'Random Text', '2018-02-01'),
   (3, 'Random Text', '2018-09-04')

I would like to return the data as below with a new column corrected date and that should always have the greatest one compared to the previous row

   (ID, TextField, DateField , Correcteddate 
   (1, 'Random Text', '2018-01-04','2018-01-04'),
   (1, 'Random Text', '2018-02-04','2018-02-04'),
   (1, 'Random Text', '2018-05-01','2018-05-01'),
   (2, 'Random Text', '2018-01-14','2018-05-01'),
   (2, 'Random Text', '2018-06-05','2018-06-05'),
   (2, 'Random Text', '2018-01-01','2018-06-05'),
   (2, 'Random Text', '2018-02-01','2018-06-05'),
   (3, 'Random Text', '2018-09-04','2018-09-04')

Tried Lead and Lag functions.

First of all, your sample data seems to be incorrect. In your input you have dates like '2017-01-14' and '2017-05-01', which seems to become '2018-01-14' and '2018-06-05' in the desired output. Same for row with ID = 3 - '2018-01-04' becomes '2018-09-04'.

Based on your output, the input looks more like:

INSERT INTO Dummy_Data (ID, TextField, DateField) VALUES
   (1, 'Random Text', '2018-01-04'),
   (1, 'Random Text', '2018-02-04'),
   (1, 'Random Text', '2018-05-01'),
   (2, 'Random Text', '2018-01-14'),
   (2, 'Random Text', '2018-06-05'),
   (2, 'Random Text', '2018-01-01'),
   (2, 'Random Text', '2018-02-01'),
   (3, 'Random Text', '2018-09-04');

Having this, you are talking about previous row but you are not defining how rows are ordered. There is no unique ID for the rows, so I suppose you are ordering them by ID and DateField, which is not the case with your input and output data. Also, there is no guarantee that the rows will be uniquely order in this way as you may have same DateField values for same ID.

Also, ask yourself, should I group the rows by ID?

And if we are ordering the rows by ID and DateField, and looking for the greatest value till now (it seems that you are not only interesting in the previous value, but you want to get the biggest value for the first two rows, then use it as output for the next row compression), you can use the following:

SELECT ID
      ,TextField
      ,DateField
      ,MAX(DateField) OVER (ORDER BY ID, DateField ASC) AS NextDate
FROM Dummy_Data