How to pick last appointment date in client list with duplicate clients

Assuming your data is in A1:D4, type this into cell E1:
=MAX((A1=$A$1:$A$4)*($D$1:$D$4))
Enter it with Ctrl-Shift-Enter
Drag that formula down to E4.

This will create a column which shows the last appointment for the patient on that row.

Though you have not provided a sample workbook so not aware of your sheet layout, just to give you an idea how you can achieve this, here is an example. You can make the changes in the formula as per your original workbook. Assuming that all your patients have the unique patient id.

	A	B	C	D	E	F	G	H	I
1	Patient ID	L Name	F Name	Event Date		Patient ID	L Name	F Name	Event Date
2	1	A	M	01/09/2014		1	A	M	08/09/2014
3	2	B	S	02/09/2014		2	B	S	07/09/2014
4	3	C	T	03/09/2014		3	C	T	09/09/2014
5	1	A	N	04/09/2014		4	D	R	06/09/2014
6	3	C	T	05/09/2014
7	4	D	R	06/09/2014
8	2	B	S	07/09/2014
9	1	A	M	08/09/2014
10	3	C	T	09/09/2014

Formulas used are as below.
In G2

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and then copy right to H2 and then down to row 5.

In I2

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and then copy down.

Glad to know that.
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One more suggestion. As you said you have 232 rows of data and col. A contains the patient ids and col. A will have repetition of patient ids as there may be more than one appointment for a particular patient. To get the summary in cols. F, G, H and I, you need to input the patients ids in col. F starting from F2. And for that you can use another formula to get the unique patient ids in col. F, so that you don't need to input them manually.
To get the unique patient ids in col. F, you may try this Array Formula in F2. Since this is an array formula so you need to confirm it with Ctrl+Shift+Enter instead of just Enter which you normally do in case of a regular excel formula. i.e. after typing the formula in the formula cell, don't press Enter but hold down the Ctrl+Shift keys together and then hit Enter. When you enter a formula as an array formula, you will notice in the formula bar that the formula gets surrounded by the Curly Brackets like {= Formula here }and if you don't see the curly brackets in the formula bar, select the formula cell and then press F2 (Function Key), hold down the Ctrl+Shift and then hit Enter to correctly enter that formula as an array formula.

In F2

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Enter this formula with Ctrl+Shift+Enter and then copy it down until you get the blank cells. (The best way is copy the above formula, go to your sheet and select F2 and then press Function Key F2, now paste the formula by Ctrl + V, hold down the Ctrl+Shift and then hit Enter and hold the fill handle and copy the formula down the rows.)

Hope this helps.