UPDATE: The New Mexico Tech tkinter website has been archived on GitHub. Also effbot was also archived since 2021.

First the best reference for Tkinter is this New Mexico Tech website. In the toc you will find a section on fonts, and in the section on Button widgets you'll find the option font.

you must have a Tkinter object to create a font

Python-2

Support for Python-2 has officially ended as of Jan 1, 2020

from Tkinter import *  # Note: UPPER case "T" in Tkinter
import tkFont
root = Tk()

Python-3

Python-3 Tk wrappers differ from Python-2

from tkinter import *  # Note: lower case "t" in tkinter
from tkinter import font as tkFont  # for convenience
root = Tk()

create a font like the example from New Mexico Tech website

helv36 = tkFont.Font(family='Helvetica', size=36, weight='bold')
# you don't have to use Helvetica or bold, this is just an example

(Note: recall for Python-3 font was imported as tkFont for convenience)

now you can set the font for button created from Button in the original post

button['font'] = helv36

The size of the button will depend on your geometry manager, EG: grid or pack. Only the grid method is covered in the layouts section by New Mexico Tech site, but effbot.org is also a great reference and he covers pack pretty well.

try:  # Python-2
    from Tkinter import *
    import tkFont
except ImportError:  # Python-3
    from tkinter import *
    from tkinter import font as tkFont
# using grid
# +------+-------------+
# | btn1 |    btn2     |
# +------+------+------+
# | btn3 | btn3 | btn4 |
# +-------------+------+
root = Tk()
# tkFont.BOLD == 'bold'
helv36 = tkFont.Font(family='Helvetica', size=36, weight=tkFont.BOLD)
btn1 = Button(text='btn1', font=helv36)
btn2 = Button(text='btn2', font=helv36)
btn3 = Button(text='btn3', font=helv36)
btn4 = Button(text='btn4', font=helv36)
btn5 = Button(text='btn5', font=helv36)
root.rowconfigure((0,1), weight=1)  # make buttons stretch when
root.columnconfigure((0,2), weight=1)  # when window is resized
btn1.grid(row=0, column=0, columnspan=1, sticky='EWNS')
btn2.grid(row=0, column=1, columnspan=2, sticky='EWNS')
btn3.grid(row=1, column=0, columnspan=1, sticky='EWNS')
btn4.grid(row=1, column=1, columnspan=1, sticky='EWNS')
btn5.grid(row=1, column=2, columnspan=1, sticky='EWNS')

Also try ttk.

Typically, when you give a button a width, that width is measured in characters (ie: width=1 means the width of one average sized character). However, if the button has an image then the width specifies a size in pixels.  

A button can contain both an image and text, so one strategy is to put a 1x1 pixel as an image so that you can specify the button size in pixels. When you do that and you change the font size, the button will not grow since it was given an absolute size.

Here is an example that illustrates the technique. Run the code, then click on "bigger" or "smaller" to see that the text changes size but the button does not.

import Tkinter as tk
import tkFont

def bigger():
    size = font.cget("size")
    font.configure(size=size+2)

def smaller():
    size = font.cget("size")
    size = max(2, size-2)
    font.configure(size=size)

root = tk.Tk()
font = tkFont.Font(family="Helvetica", size=12)

toolbar = tk.Frame(root)
container = tk.Frame(root)

toolbar.pack(side="top", fill="x")
container.pack(side="top", fill="both", expand=True)

bigger = tk.Button(toolbar, text="Bigger", command=bigger)
smaller = tk.Button(toolbar, text="Smaller", command=smaller)

bigger.pack(side="left")
smaller.pack(side="left")

pixel = tk.PhotoImage(width=1, height=1)
for row in range(3):
    container.grid_rowconfigure(row, weight=1)
    for column in range(3):
        container.grid_columnconfigure(column, weight=1)
        button = tk.Button(container, font=font, text="x",
            image=pixel, compound="center", width=20, height=20)
        button.grid(row=row, column=column)

root.mainloop()

All of that being said, there is almost never a time when this is a good idea. If the user wants a larger font, the whole UI should adapt. Tkinter is really good at making that happen, to the point where it all mostly works by default.